2018 Winter Olympics opening ceremony - wikipedia 2018 Winter Olympics opening ceremony Jump to : navigation , search This article is incomplete . Please help to improve it , or discuss the issue on the talk page . ( February 2018 ) 2018 Winter Olympics opening ceremony Yuna Kim , South Korean Olympic figure skating champion , lights the Olympic cauldron Date 9 February 2018 ( 2018 - 02 - 09 ) Time 20 : 00 -- 22 : 20 KST ( UTC + 9 ) Location Pyeongchang , South Korea Coordinates 37 ° 40 ′ 03 '' N 128 ° 42 ′ 20 '' E  /  37.66750 ° N 128.70556 ° E  / 37.66750 ; 128.70556 Coordinates : 37 ° 40 ′ 03 '' N 128 ° 42 ′ 20 '' E  /  37.66750 ° N 128.70556 ° E  / 37.66750 ; 128.70556 Also known as Peace in Motion Filmed by SBS and OBS Part of a series on the 2018 Winter Olympics Bid process ( bid details ) Venues Torch relay Mascots Broadcasters Opening ceremony ( flag bearers ) Chronological summary Medal table ( medalists ) World and Olympic records Controversies ( ban on Russian athletes ) Closing ceremony ( flag bearers ) Paralympics IOC KOC POCOG The opening ceremony of the 2018 Winter Olympics was held at the Pyeongchang Olympic Stadium in Pyeongchang , South Korea on 9 February 2018 . It began at 20 : 00 KST and finished at approximately 22 : 20 KST . The Games were officially opened by President of the Republic of Korea Moon Jae - in . Contents ( hide ) 1 Preparations 2 Ceremony 3 Programme 3.1 Parade of Nations 3.2 Torch lighting 3.3 Wish Fire 3.4 Intel ® Drone Light Show 4 Controversies 4.1 Mike Pence 4.2 Russian hacking 5 Dignitaries in attendance 6 Anthems 7 Ceremony key team 8 References 9 External links Preparations ( edit ) The site of the opening ceremony , Pyeongchang Olympic Stadium , was built specifically for the Games . The pentagonal stadium seats 35,000 . The organizers for the event said the shape was chosen because it is a combination of different shapes , a circle , a square , and a triangle , which represent heaven , earth , and mankind . No Olympic or Paralympic events will be held at the stadium , which will only be used for the opening and closing ceremonies . The venue will be torn down afterwards . The broadcast of the Opening Ceremony of the 2018 Winter Olympic Games was available in more than 200 countries around the world . Ha Hyun - woo of the band Guckkasten , Ahn Ji - young of the musical duo Bolbbalgan4 , Lee Eun - mi and Jeon In - kwon with his band Deulgukhwa performed and sang John Lennon 's `` Imagine '' at the opening ceremony . The Rainbow Children 's Choir performed an arrangement , with Korean traditional musical instruments , of the national anthem , `` Aegukga '' . The multiethnic children 's choir was formed to educate the Korean public about multicultural families living in Korea . Insooni sang the theme song for the PyeongChang 2018 Olympic Torch Relay , `` Let Everyone Shine '' . Ceremony ( edit ) The ceremony 's message centered on peace , passion , harmony , and convergence . Five children from rural Gangwon province led the ceremony , which included one Inmyeonjo ( a mythical creature with a body of a bird and a head of a man , a creature that only appears when the world is at peace ) , four mythical creatures , including a white tiger -- spirit animal protector of Korea and the mascot of these Games , natural floral and fauna , and a cast of 2,000 . Five children were used to symbolize the five Olympic rings , and the five names were chosen to represent fire , water , wood , metal , and earth , the five elements that are believed to make up the Earth . Augmented reality and 5G technology were also incorporated in the event . The largest drone show in history , featuring 1,218 Shooting Star drones , was planned for the ceremony but cancelled at the last minute ; television audiences were shown a version that had been recorded the previous December . During the opening ceremony the organizing committee was a victim of a cyber attack , but without major consequences . Programme ( edit ) Parade of Nations ( edit ) Main article : 2018 Winter Olympics Parade of Nations The Parade of Nations was led , according to custom due to hosting the original ancient Olympics , by the Greek team , followed by other competing countries in alphabetical order based on their names in the Korean language , with the host country , South Korea , concluding the march . The delegations from both the host nation South Korea and North Korea ( Korea at the 2018 Winter Olympics ) marched under the Korean unification flag . South Korean Won Yun - jong and North Korean Hwang Chung - gum both held the flagpole . Even though the temperature in the stadium during the ceremony was very cold , Bermuda 's delegation wore shorts and Tongan Pita Taufatofua repeated his shirtless and oiled up Summer Olympic appearance . Background music for the parade began with an instrumental version of `` Hand in Hand '' by Koreana and included modern dance remixes of songs such as `` 단발 머리 '' by Cho Yong - pil and of traditional Korean music , as well as the following K - pop hits : `` Gangnam Style '' by Psy , `` Likey '' by Twice , `` Fantastic Baby '' by Big Bang , `` DNA '' by BTS , and `` Red Flavor '' by Red Velvet . `` Red Flavor '' was cut off just as the unified Korean team was introduced . Torch lighting ( edit ) See also : 2018 Winter Olympics torch relay Cauldron The Olympic flame was on a 101 - day relay across South Korea . The start of the ceremony takes place in Olympia of Greece . Apostles Angelis , 24 years old cross-country skier , is selected to be the first torchbearer . He said `` It is a great honour for me to be chosen as the first torchbearer for the Olympic Winter Games of 2018 . It is truly a unique moment that I am looking forward to . I feel very proud and with a unique sense of happiness . '' Park Ji - Sung , football player in Republic of Korea , was the second person to hold the torch . After the flame touring around Greece for a week , it reached the Acropolis on 30th of October . There were 36 ceremonies held in different cities over a week , then the flame arrived to the PyeongChang Panathenaic Stadium . More than 500 torchbearers participated to follow the rituals of Olympic cycle , and carried the message of peace . In the stadium , Opera soloist Sumi Hwang sang the Olympic anthem during the torch lighting ceremony . The final person to hold the torch was Inbee Park , the golfer from Republic of Korea . Chung Su - hyon from North Korea and Park Jong - ah from South Korea carried the torch and headed up stairs toward the cauldron . As the torch neared the cauldron , an ice skating rink with South Korean figure skater Yuna Kim was revealed . Yuna Kim is the well - known figure skater who won gold medal in 2010 and silver metal in 2014 . The cauldron was designed to represent a full moon when it is not lit . Kim received the torch and lit the cauldron . Wish fire ( edit ) The final segment entitled `` Wish Fire '' featured Korean dance group Just Jerk ( who became known internationally for their participation in the twelfth season of America 's Got Talent ) entered the stage dressed as Dokkaebi , and performers with sparklers on roller blades . This segment in total featured 2,000 fireworks within and out of the stadium , some reaching as high as 1,300 metres ( 4,300 ft ) . Intel ® drone light Show ( edit ) More than 1,200 Intel Unmanned Aerial Vehicles ( UAVs ) or drones flying above Pyeong Chang to celebrate the opening of 2018 Winter Olympic Games . Intel ® drone team now holds the Guinness World Record of flying largest number of drones . The drones flew together to form custom animations illustrating snow boarding , skiing , other different sports , as well as the iconic Olympic rings . The Shooting Star drones carried on the symbol of unity and progress from lighting of the torch . Controversies ( edit ) Mike Pence ( edit ) While there is no protocol requiring guests to stand for the host country 's arrival , United States Vice President Mike Pence was criticized for not standing , as North and South Korean athletes entered during the Parade of Athletes . Pence was accused of hypocrisy , with critics drawing parallels to national anthem protests by NFL athletes . Russian hacking ( edit ) On 24 February , the Washington Post reported that U.S. intelligence uncovered Russian spies hacked computers during the Opening Ceremony , an act that was previously speculated to be the work of North Korea . Analysts believed the Russians instigated the February 9 attack as a way to retaliate for the International Olympic Committee 's decision to ban Russia from the 2018 Winter Olympics as punishment for doping violations . Dignitaries in attendance ( edit ) Mike Pence , Vice President of the United States Co-Prince of Andorra and President of France Emmanuel Macron Governor General Julie Payette Politburo standing member Han Zheng Vice President Óscar Naranjo President Kersti Kaljulaid Minister of Europe and Foreign Affairs Jean - Yves Le Drian and Minister of Sports Laura Flessel President Frank - Walter Steinmeier Sports Minister Luca Lotti Prime Minister Shinzo Abe President of Korea Moon Jae - In President Dalia Grybauskaitė President Hashim Thaçi Grand Duke Henri Prince Albert II King Willem Alexander , Queen Máxima and Prime Minister Mark Rutte President of the Presidium of the Supreme People 's Assembly Kim Yong - nam and Director of the Propaganda and Agitation Department of the Workers ' Party of Korea Kim Yo - jong President Andrzej Duda Minister of Youth and Sports Vanja Udovičić President Andrej Kiska President Alain Berset Princess Anne Vice President Mike Pence and Second Lady Karen Pence Secretary - General António Guterres Kim Yo - jong attending the ceremony On 9 February 2018 , Kim Yo - jong -- sister of Kim Jong - un -- attended the ceremony in Pyeongchang , South Korea . This was a first time that a member of the ruling Kim dynasty had visited South Korea since the Korean War . Kim Yo - jong shook hands with South Korean president Moon Jae - in before sitting down to watch the ceremony together . Anthems ( edit ) Rainbow Children 's Choir -- South Korean National Anthem Sumi Hwang -- Olympic Hymn Ceremony key team ( edit ) Artistic Director : Yang Jung - woong Executive Director , creative : Song Seung - whan Executive Director , cinematography : Lim Tae - hong , Andrew References ( edit ) ^ Jump up to : Chappell , Bill ( February 9 , 2018 ) . `` Winter Olympics Opening Ceremony : Pyeongchang Welcomes The World '' . NPR . Retrieved February 11 , 2018 . Jump up ^ Eaton , Joe ( February 9 , 2018 ) . `` It 's Time for the Pop - Up Olympics '' . CITYLAB . Retrieved February 11 , 2018 . Jump up ^ `` Pyeongchang 2018 Opening Ceremony LIVE on 9 February 2018 '' . 18 January 2018 . Retrieved 8 February 2018 . ^ Jump up to : Hong , C. ( 30 January 2018 ) . `` Ha Hyun Woo , Bolbbalgan4 's Ahn Ji Young , And More To Perform At Olympics Opening Ceremony '' . Soompi . Viki , Inc . Retrieved 1 February 2018 . ^ Jump up to : Garcia , Maira ( 10 February 2018 ) . `` Who Were the Singers at the Opening Ceremony of the Winter Olympics ? '' . nytimes.com . Retrieved 2018 - 02 - 11 . Jump up ^ Garcia , Maira ( 10 Feb 2018 ) . `` Who Were the Singers at the Opening Ceremony of the Winter Olympics ? '' . PyeongChang 2018 . Jump up ^ PyeongChang 2018 ( 21 September 2017 ) . `` ( ENG ) Theme song for the PyeongChang 2018 Olympic Torch Relay ( ENG sub ) '' . Retrieved 8 February 2018 -- via YouTube . Jump up ^ `` Inside the Olympics Opening Ceremony World - Record Drone Show '' . WIRED . Retrieved 2018 - 02 - 09 . ^ Jump up to : `` Winter Olympics 2018 opening ceremony -- live ! '' . Guardian . 9 February 2018 . Retrieved 9 February 2018 . ^ Jump up to : `` Winter Olympics 2018 opening ceremony review : Pyeongchang unites the world in a blizzard of emotion and effects '' . The Telegraph . Retrieved February 11 , 2018 . Jump up ^ Kaufman , Gil ( 6 December 2017 ) . `` 2018 Winter Olympics Kick Off With Korean Stars PSY & BTS During Opening Ceremonies '' . Billboard . Retrieved 9 February 2018 . ^ Jump up to : `` Olympia Awaits Flame Lighting Ceremony for PyeongChang 2018 '' . Olympic Games . 19 October 2017 . Jump up ^ `` Information to the Media - PyeongChang 2018 Olympic Flame - Lighting Ceremony '' . Olympic Games . 23 October 2017 . ^ Jump up to : Keh , Andrew ( 9 Feb 2018 ) . `` Winter Olympics 2018 Opening Ceremony : Highlights and Analysis '' . PyeongChang 2018 . Jump up ^ Behind the Scenes of the PyeongChang 2018 Opening Ceremony Jump up ^ `` Intel ® Drones at The Olympic Games '' . intel ® . Jump up ^ `` A Record - Breaking Show at PyeongChang 2018 '' . intel ® . 2018 . ^ Jump up to : Radnofsky , Louise ; Cheng , Jonathan ( 10 February 2018 ) . `` Pence Sat During Opening Ceremony . Some Koreans Took Offense '' . The Wall Street Journal . Jump up ^ Banks , Paul M. ( 9 February 2018 ) . `` Winter Olympics Verify Mike Pence Hypocrisy on NFL National Anthem Protests '' . ChicagoNow . Retrieved 13 February 2018 . Jump up ^ Zilber , Ariel ( 10 February 2018 ) . `` Mike Pence accused of hypocrisy for refusing to stand for Unified Korea during Winter Olympics opening ceremony -- months after walking out of an NFL game because of kneeling players '' . Daily Mail . Retrieved 13 February 2018 . Jump up ^ Meeks , David ( 12 February 2018 ) . `` By not standing for Korea at the 2018 Winter Olympics , VP Mike Pence embarrassed America '' . USA Today . Retrieved 13 February 2018 . Jump up ^ Report : Russian spies behind Winter Olympics cyber attack . US Today . Retrieved 26 February 2018 . Jump up ^ Russian spies reportedly hacked Winter Olympics . CNET . Retrieved 26 February 2018 . ^ Jump up to : `` Winter Olympics opening ceremony : Koreans enter under unified flag -- as it happened '' . The Guardian . Retrieved February 11 , 2018 . Jump up ^ `` Gov. Gen. Julie Payette to attend opening of winter Olympics '' . Jump up ^ `` SOUTH KOREA - CHINA - HAN ZHENG - WINTER OLYMPICS - OPENING CEREMONY '' . February 9 , 2018 . Retrieved February 10 , 2018 . Jump up ^ `` Korea - Estonia summit to help launch peace through Olympics '' . Korea.net. 7 February 2018 . Retrieved 10 February 2018 . President Moon Jae - in held a meeting with President Kersti Kaljulaid of Estonia who is on a visit to Korea to attend the PyeongChang 2018 Olympic and Paralympic Winter Games . Jump up ^ `` 평창 오는 외국 정상 벌써 15 명 ... ' 올림픽 외교 ' 꽃 피운다 '' . 26 December 2017 . Retrieved 8 February 2018 . Jump up ^ `` Presidente de Alemania apoya diálogo entre Coreas '' ( in Spanish ) . Deutsche Welle . Retrieved 10 February 2018 . Jump up ^ Daniela Cotto ( 9 February 2018 ) . `` Il ministro Lotti a Casa Italia , l'incontro con Lindsey Vonn '' ( in Italian ) . La Stampa . Retrieved 10 February 2018 . Jump up ^ `` Mandatarios de Japón y Corea del Sur acuerdan continuar ejerciendo presión sobre Corea del Norte '' ( in Spanish ) . NHK World . Retrieved 10 February 2018 . Abe y Moon sostuvieron una reunión de aproximadamente una hora en un hotel cerca del Estadio Olímpico de Pyeongchang antes de asistir el viernes a la ceremonia de apertura de los Juegos Olímpicos de Invierno . Jump up ^ `` Lío de protocolo ante la inauguración de los juegos de invierno de Pyeongchang '' ( in Spanish ) . La Vanguardia . Retrieved 10 February 2018 . Jump up ^ `` The President attends the opening ceremony of Winter Olympics '' . President of the Republic of Lithuania. 9 February 2018 . Retrieved 10 February 2018 . Jump up ^ Alan Crosby ( 10 February 2018 ) . `` Winter Olympics Debut A 10th - Anniversary Gift For Kosovo '' . Radio Free Europe / Radio Liberty . Retrieved 10 February 2018 . Hashim Thaci , president of Kosovo , said on February 5 during a ceremony presenting the official Kosovar flag to Tahiri . Jump up ^ `` Grand Duke Henri to Attend Winter Olympics '' . Chronicle.lu . Retrieved 10 February 2018 . Jump up ^ `` Prince Albert II of Monaco ( C ) reacts as he visits the House of Switzerland during the PyeongChang 2018 Olympic Games , South Korea '' . european pressphoto agency . Retrieved 10 February 2018 . Jump up ^ `` Koningspaar en premier Rutte bij openingsceremonie Pyeongchang '' . Jump up ^ Talmadge , Eric ( 9 February 2018 ) . `` At Olympic Games , Kim Jong Un 's sister takes VIP seat '' . ABC News . AP . Retrieved 2018 - 02 - 09 . Jump up ^ `` Szczerski o rozmowie Dudy z wiceprezydentem USA : od razu wiadomo , że się odbędzie '' . Retrieved 8 February 2018 . Jump up ^ `` Зимске олимпијске игре свечано отворене у Пјончангу -- СПОРТСКИ ЖУРНАЛ '' . Jump up ^ `` 33 개국 정상급 45 명 참석 밝혀 ... 靑 경호 처가 근접 보호 '' . Retrieved 8 February 2018 . Jump up ^ `` JO 2018 : Alain Berset à Pyeongchang pour affirmer la présence de la Suisse '' ( in French ) . La Côte. 10 February 2018 . Retrieved 10 February 2018 . Jump up ^ `` Guterres se reunió con el líder norcoreano Kim Yong - nam '' ( in Spanish ) . Deutsche Welle. 10 February 2018 . Retrieved 10 February 2018 . El secretario general de la ONU , Antonio Guterres , se encontró con el presidente nominal de Corea del Norte , Kim Yong - nam , en el marco de la inauguración de los Juegos Olímpicos de invierno en Pyeongchang . Jump up ^ `` 美 펜스 , 올림픽 대표단 끌고 평창 行 ... 이방 카 참석 은 ' 미정 ' '' . 10 January 2018 . Retrieved 8 February 2018 . Jump up ^ `` 9 February 2018 , Buckingham Palace '' . Court Circular. 9 February 2018 . Retrieved 21 February 2018 . The Princess Royal , Member , International Olympic Committee , and President , British Olympic Association , this evening attended the Opening Reception and Ceremony of the XXIII Winter Olympic Games . Jump up ^ Haas , Benjamin ( 9 February 2018 ) . `` US vice-president skips Olympics dinner in snub to North Korea officials '' . the Guardian . Retrieved 2018 - 02 - 09 . Jump up ^ `` Olympics - Pyeongchang picks ' Nanta ' producer for 2018 ceremonies '' . Seoul , South Korea . Reuters. 2 July 2015 . Retrieved 1 February 2018 . External links ( edit ) Media related to 2018 Winter Olympics opening ceremony at Wikimedia Commons ( hide ) Olympic Games ceremonies Opening ceremonies Summer 1896 1900 1904 1908 1912 1920 1924 1928 1932 1936 1948 1952 1956 1960 1964 1968 1972 1976 1980 1984 1988 1992 2000 2008 2012 2016 2020 2024 2028 2032 Winter 1924 1928 1932 1936 1948 1952 1956 1960 1964 1968 1972 1976 1980 1984 1988 1992 1994 1998 2002 2006 2014 2018 2022 2026 2030 Closing ceremonies Summer 1896 1900 1904 1908 1912 1920 1924 1928 1932 1936 1948 1952 1956 1960 1964 1968 1972 1976 1980 1984 1988 1992 2000 2008 2012 2016 2020 2024 2028 2032 Winter 1924 1928 1932 1936 1948 1952 1956 1960 1964 1968 1972 1976 1980 1984 1988 1992 1994 1998 2002 2006 2014 2018 2022 2026 2030 Olympics portal South Korea portal Retrieved from `` https://en.wikipedia.org/w/index.php?title=2018_Winter_Olympics_opening_ceremony&oldid=836009151 '' Categories : 2018 Winter Olympics Ceremonies in South Korea Olympics opening ceremonies Hidden categories : CS1 Spanish - language sources ( es ) CS1 Italian - language sources ( it ) CS1 French - language sources ( fr ) Articles to be expanded from February 2018 Use dmy dates from February 2018 Use British English from February 2018 Coordinates on Wikidata Talk Contents About Wikipedia Asturianu Español Français 한국어 日本 語 ไทย Українська 中文 Edit links This page was last edited on 12 April 2018 , at 03 : 39 . 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Angle - wikipedia Angle Jump to : navigation , search Not to be confused with Angel . This article is about angles in geometry . For other uses , see Angle ( disambiguation ) . Basic angle types Types of angles 2D angles Right Interior Exterior 2D angle pairs Adjacent Vertical Complementary Supplementary Transversal 3D angles Dihedral An angle formed by two rays emanating from a vertex . In planar geometry , an angle is the figure formed by two rays , called the sides of the angle , sharing a common endpoint , called the vertex of the angle . Angles formed by two rays lie in a plane , but this plane does not have to be a Euclidean plane . Angles are also formed by the intersection of two planes in Euclidean and other spaces . These are called dihedral angles . Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection . Similar statements hold in space , for example , the spherical angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles . Angle is also used to designate the measure of an angle or of a rotation . This measure is the ratio of the length of a circular arc to its radius . In the case of a geometric angle , the arc is centered at the vertex and delimited by the sides . In the case of a rotation , the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation . The word angle comes from the Latin word angulus , meaning `` corner '' ; cognate words are the Greek ἀγκύλος ( ankylοs ) , meaning `` crooked , curved , '' and the English word `` ankle '' . Both are connected with the Proto - Indo - European root * ank - , meaning `` to bend '' or `` bow '' . Euclid defines a plane angle as the inclination to each other , in a plane , of two lines which meet each other , and do not lie straight with respect to each other . According to Proclus an angle must be either a quality or a quantity , or a relationship . The first concept was used by Eudemus , who regarded an angle as a deviation from a straight line ; the second by Carpus of Antioch , who regarded it as the interval or space between the intersecting lines ; Euclid adopted the third concept , although his definitions of right , acute , and obtuse angles are certainly quantitative . Contents ( hide ) 1 Identifying angles 2 Types of angles 2.1 Individual angles 2.2 Equivalence angle pairs 2.3 Vertical and adjacent angle pairs 2.4 Combining angle pairs 2.5 Polygon related angles 2.6 Plane related angles 3 Measuring angles 3.1 Angle addition postulate 3.2 Units 3.3 Positive and negative angles 3.4 Alternative ways of measuring the size of an angle 3.5 Astronomical approximations 4 Angles between curves 5 Bisecting and trisecting angles 6 Dot product and generalisations 6.1 Inner product 6.2 Angles between subspaces 6.3 Angles in Riemannian geometry 6.4 Hyperbolic angle 7 Angles in geography and astronomy 8 See also 9 Notes 10 References 11 External links Identifying angles ( edit ) In mathematical expressions , it is common to use Greek letters ( α , β , γ , θ , φ , ... ) to serve as variables standing for the size of some angle . ( To avoid confusion with its other meaning , the symbol π is typically not used for this purpose . ) Lower case Roman letters ( a , b , c , ... ) are also used , as are upper case Roman letters in the context of polygons . See the figures in this article for examples . In geometric figures , angles may also be identified by the labels attached to the three points that define them . For example , the angle at vertex A enclosed by the rays AB and AC ( i.e. the lines from point A to point B and point A to point C ) is denoted ∠ BAC ( in Unicode U + 2220 ∠ ANGLE ) or B A C ^ ( \ displaystyle ( \ widehat ( \ rm ( BAC ) ) ) ) . Sometimes , where there is no risk of confusion , the angle may be referred to simply by its vertex ( `` angle A '' ) . Potentially , an angle denoted , say , ∠ BAC might refer to any of four angles : the clockwise angle from B to C , the anticlockwise angle from B to C , the clockwise angle from C to B , or the anticlockwise angle from C to B , where the direction in which the angle is measured determines its sign ( see Positive and negative angles ) . However , in many geometrical situations it is obvious from context that the positive angle less than or equal to 180 degrees is meant , and no ambiguity arises . Otherwise , a convention may be adopted so that ∠ BAC always refers to the anticlockwise ( positive ) angle from B to C , and ∠ CAB to the anticlockwise ( positive ) angle from C to B . Types of angles ( edit ) `` Oblique angle '' redirects here . For the cinematographic technique , see Dutch angle . Individual angles ( edit ) Right angle . Reflex angle . Acute ( a ) , obtuse ( b ) , and straight ( c ) angles . The acute and obtuse angles are also known as oblique angles . Angles smaller than a right angle ( less than 90 ° ) are called acute angles ( `` acute '' meaning `` sharp '' ) . An angle equal to 1 / 4 turn ( 90 ° or π / 2 radians ) is called a right angle . Two lines that form a right angle are said to be normal , orthogonal , or perpendicular . Angles larger than a right angle and smaller than a straight angle ( between 90 ° and 180 ° ) are called obtuse angles ( `` obtuse '' meaning `` blunt '' ) . An angle equal to 1 / 2 turn ( 180 ° or π radians ) is called a straight angle . Angles larger than a straight angle but less than 1 turn ( between 180 ° and 360 ° ) are called reflex angles . An angle equal to 1 turn ( 360 ° or 2π radians ) is called a full angle , complete angle , or a perigon . Angles that are not right angles or a multiple of a right angle are called oblique angles . The names , intervals , and measured units are shown in a table below : Name acute right angle obtuse straight reflex perigon Units Interval Turns ( 0 , 1 / 4 ) 1 / 4 ( 1 / 4 , 1 / 2 ) 1 / 2 ( 1 / 2 , 1 ) Radians ( 0 , 1 / 2π ) 1 / 2π ( 1 / 2π , π ) π ( π , 2π ) 2π Degrees ( 0 , 90 ) ° 90 ° ( 90 , 180 ) ° 180 ° ( 180 , 360 ) ° 360 ° Gons ( 0 , 100 ) 100 ( 100 , 200 ) 200 ( 200 , 400 ) 400 Equivalence angle pairs ( edit ) Angles that have the same measure ( i.e. the same magnitude ) are said to be equal or congruent . An angle is defined by its measure and is not dependent upon the lengths of the sides of the angle ( e.g. all right angles are equal in measure ) . Two angles which share terminal sides , but differ in size by an integer multiple of a turn , are called coterminal angles . A reference angle is the acute version of any angle determined by repeatedly subtracting or adding straight angle ( 1 / 2 turn , 180 ° , or π radians ) , to the results as necessary , until the magnitude of result is an acute angle , a value between 0 and 1 / 4 turn , 90 ° , or π / 2 radians . For example , an angle of 30 degrees has a reference angle of 30 degrees , and an angle of 150 degrees also has a reference angle of 30 degrees ( 180 -- 150 ) . An angle of 750 degrees has a reference angle of 30 degrees ( 750 -- 720 ) . Vertical and adjacent angle pairs ( edit ) Angles A and B are a pair of vertical angles ; angles C and D are a pair of vertical angles . When two straight lines intersect at a point , four angles are formed . Pairwise these angles are named according to their location relative to each other . A pair of angles opposite each other , formed by two intersecting straight lines that form an `` X '' - like shape , are called vertical angles or opposite angles or vertically opposite angles . They are abbreviated as vert . opp . ∠ s . The equality of vertically opposite angles is called the vertical angle theorem . Eudemus of Rhodes attributed the proof to Thales of Miletus . The proposition showed that since both of a pair of vertical angles are supplementary to both of the adjacent angles , the vertical angles are equal in measure . According to a historical Note , when Thales visited Egypt , he observed that whenever the Egyptians drew two intersecting lines , they would measure the vertical angles to make sure that they were equal . Thales concluded that one could prove that all vertical angles are equal if one accepted some general notions such as : all straight angles are equal , equals added to equals are equal , and equals subtracted from equals are equal . In the figure , assume the measure of Angle A = x . When two adjacent angles form a straight line , they are supplementary . Therefore , the measure of Angle C = 180 − x . Similarly , the measure of Angle D = 180 − x . Both Angle C and Angle D have measures equal to 180 − x and are congruent . Since Angle B is supplementary to both Angles C and D , either of these angle measures may be used to determine the measure of Angle B. Using the measure of either Angle C or Angle D we find the measure of Angle B = 180 − ( 180 − x ) = 180 − 180 + x = x . Therefore , both Angle A and Angle B have measures equal to x and are equal in measure . Angles A and B are adjacent . Adjacent angles , often abbreviated as adj . ∠ s , are angles that share a common vertex and edge but do not share any interior points . In other words , they are angles that are side by side , or adjacent , sharing an `` arm '' . Adjacent angles which sum to a right angle , straight angle or full angle are special and are respectively called complementary , supplementary and explementary angles ( see `` Combine angle pairs '' below ) . A transversal is a line that intersects a pair of ( often parallel ) lines and is associated with alternate interior angles , corresponding angles , interior angles , and exterior angles . Combining angle pairs ( edit ) There are three special angle pairs which involve the summation of angles : Complementary angles are angle pairs whose measures sum to one right angle ( 1 / 4 turn , 90 ° , or π / 2 radians ) . If the two complementary angles are adjacent their non-shared sides form a right angle . In Euclidean geometry , the two acute angles in a right triangle are complementary , because the sum of internal angles of a triangle is 180 degrees , and the right angle itself accounts for ninety degrees . The adjective complementary is from Latin complementum , associated with the verb complere , `` to fill up '' . An acute angle is `` filled up '' by its complement to form a right angle . The difference between an angle and a right angle is termed the complement of the angle . If angles A and B are complementary , the following relationships hold : sin 2  A + sin 2  B = 1 cos 2  A + cos 2  B = 1 tan  A = cot  B sec  A = csc  B ( \ displaystyle ( \ begin ( aligned ) & \ sin ^ ( 2 ) A+ \ sin ^ ( 2 ) B = 1&& \ cos ^ ( 2 ) A+ \ cos ^ ( 2 ) B = 1 \ \ ( 3pt ) & \ tan A = \ cot B&& \ sec A = \ csc B \ end ( aligned ) ) ) ( The tangent of an angle equals the cotangent of its complement and its secant equals the cosecant of its complement . ) The prefix `` co - '' in the names of some trigonometric ratios refers to the word `` complementary '' . Two angles that sum to a straight angle ( 1 / 2 turn , 180 ° , or π radians ) are called supplementary angles . If the two supplementary angles are adjacent ( i.e. have a common vertex and share just one side ) , their non-shared sides form a straight line . Such angles are called a linear pair of angles . However , supplementary angles do not have to be on the same line , and can be separated in space . For example , adjacent angles of a parallelogram are supplementary , and opposite angles of a cyclic quadrilateral ( one whose vertices all fall on a single circle ) are supplementary . If a point P is exterior to a circle with center O , and if the tangent lines from P touch the circle at points T and Q , then ∠ TPQ and ∠ TOQ are supplementary . The sines of supplementary angles are equal . Their cosines and tangents ( unless undefined ) are equal in magnitude but have opposite signs . In Euclidean geometry , any sum of two angles in a triangle is supplementary to the third , because the sum of internal angles of a triangle is a straight angle . Two angles that sum to a complete angle ( 1 turn , 360 ° , or 2π radians ) are called explementary angles or conjugate angles . The difference between an angle and a complete angle is termed the explement of the angle or conjugate of an angle . The complementary angles a and b ( b is the complement of a , and a is the complement of b ) . A reflex angle and its conjugate are explementary angles , and their sum is a complete angle . The angles a and b are supplementary angles . Polygon related angles ( edit ) Internal and external angles . An angle that is part of a simple polygon is called an interior angle if it lies on the inside of that simple polygon . A simple concave polygon has at least one interior angle that is a reflex angle . In Euclidean geometry , the measures of the interior angles of a triangle add up to π radians , 180 ° , or 1 / 2 turn ; the measures of the interior angles of a simple convex quadrilateral add up to 2π radians , 360 ° , or 1 turn . In general , the measures of the interior angles of a simple convex polygon with n sides add up to ( n − 2 ) π radians , or 180 ( n − 2 ) degrees , ( 2n − 4 ) right angles , or ( n / 2 − 1 ) turn . The supplement of an interior angle is called an exterior angle , that is , an interior angle and an exterior angle form a linear pair of angles . There are two exterior angles at each vertex of the polygon , each determined by extending one of the two sides of the polygon that meet at the vertex ; these two angles are vertical angles and hence are equal . An exterior angle measures the amount of rotation one has to make at a vertex to trace out the polygon . If the corresponding interior angle is a reflex angle , the exterior angle should be considered negative . Even in a non-simple polygon it may be possible to define the exterior angle , but one will have to pick an orientation of the plane ( or surface ) to decide the sign of the exterior angle measure . In Euclidean geometry , the sum of the exterior angles of a simple convex polygon will be one full turn ( 360 ° ) . The exterior angle here could be called a supplementary exterior angle . Exterior angles are commonly used in Logo Turtle Geometry when drawing regular polygons . In a triangle , the bisectors of two exterior angles and the bisector of the other interior angle are concurrent ( meet at a single point ) . In a triangle , three intersection points , each of an external angle bisector with the opposite extended side , are collinear . In a triangle , three intersection points , two of them between an interior angle bisector and the opposite side , and the third between the other exterior angle bisector and the opposite side extended , are collinear . Some authors use the name exterior angle of a simple polygon to simply mean the explement exterior angle ( not supplement ! ) of the interior angle . This conflicts with the above usage . Plane related angles ( edit ) The angle between two planes ( such as two adjacent faces of a polyhedron ) is called a dihedral angle . It may be defined as the acute angle between two lines normal to the planes . The angle between a plane and an intersecting straight line is equal to ninety degrees minus the angle between the intersecting line and the line that goes through the point of intersection and is normal to the plane . Measuring angles ( edit ) The size of a geometric angle is usually characterized by the magnitude of the smallest rotation that maps one of the rays into the other . Angles that have the same size are said to be equal or congruent or equal in measure . In some contexts , such as identifying a point on a circle or describing the orientation of an object in two dimensions relative to a reference orientation , angles that differ by an exact multiple of a full turn are effectively equivalent . In other contexts , such as identifying a point on a spiral curve or describing the cumulative rotation of an object in two dimensions relative to a reference orientation , angles that differ by a non-zero multiple of a full turn are not equivalent . The measure of angle θ ( in radians ) is the quotient of s and r . In order to measure an angle θ , a circular arc centered at the vertex of the angle is drawn , e.g. with a pair of compasses . The ratio of the length s of the arc by the radius r of the circle is the measure of the angle in radians . The measure of the angle in another angular unit is then obtained by multiplying its measure in radians by the scaling factor k / 2π , where k is the measure of a complete turn in the chosen unit ( for example 360 for degrees or 400 for gradians ) : θ = k s 2 π r . ( \ displaystyle \ theta = k ( \ frac ( s ) ( 2 \ pi r ) ) . ) The value of θ thus defined is independent of the size of the circle : if the length of the radius is changed then the arc length changes in the same proportion , so the ratio s / r is unaltered . ( Proof . The formula above can be rewritten as k = θr / s . One turn , for which θ = n units , corresponds to an arc equal in length to the circle 's circumference , which is 2πr , so s = 2πr . Substituting n for θ and 2πr for s in the formula , results in k = nr / 2πr = n / 2π . ) Angle addition postulate ( edit ) The angle addition postulate states that if B is in the interior of angle AOC , then m ∠ A O C = m ∠ A O B + m ∠ B O C ( \ displaystyle m \ angle AOC = m \ angle AOB + m \ angle BOC ) The measure of the angle AOC is the sum of the measure of angle AOB and the measure of angle BOC . In this postulate it does not matter in which unit the angle is measured as long as each angle is measured in the same unit . Units ( edit ) See also : Angular unit Units used to represent angles are listed below in descending magnitude order . Of these units , the degree and the radian are by far the most commonly used . Angles expressed in radians are dimensionless for the purposes of dimensional analysis . Most units of angular measurement are defined such that one turn ( i.e. one full circle ) is equal to n units , for some whole number n . The two exceptions are the radian and the diameter part . Turn ( n = 1 ) The turn , also cycle , full circle , revolution , and rotation , is complete circular movement or measure ( as to return to the same point ) with circle or ellipse . A turn is abbreviated τ , cyc , rev , or rot depending on the application , but in the acronym rpm ( revolutions per minute ) , just r is used . A turn of n units is obtained by setting k = 1 / 2π in the formula above . The equivalence of 1 turn is 360 ° , 2π rad , 400 grad , and 4 right angles . The symbol τ can also be used as a mathematical constant to represent 2π radians . Used in this way ( k = τ / 2π ) allows for radians to be expressed as a fraction of a turn . For example , half a turn is τ / 2 = π . Quadrant ( n = 4 ) The quadrant is 1 / 4 of a turn , i.e. a right angle . It is the unit used in Euclid 's Elements. 1 quad. = 90 ° = π / 2 rad = 1 / 4 turn = 100 grad . In German the symbol has been used to denote a quadrant . Sextant ( n = 6 ) The sextant ( angle of the equilateral triangle ) is 1 / 6 of a turn . It was the unit used by the Babylonians , and is especially easy to construct with ruler and compasses . The degree , minute of arc and second of arc are sexagesimal subunits of the Babylonian unit . 1 Babylonian unit = 60 ° = π / 3 rad ≈ 1.047197551 rad . θ = s / r rad = 1 rad . Radian ( n = 2π = 6.283 ... ) The radian is the angle subtended by an arc of a circle that has the same length as the circle 's radius . The case of radian for the formula given earlier , a radian of n = 2π units is obtained by setting k = 2π / 2π = 1 . One turn is 2π radians , and one radian is 180 / π degrees , or about 57.2958 degrees . The radian is abbreviated rad , though this symbol is often omitted in mathematical texts , where radians are assumed unless specified otherwise . When radians are used angles are considered as dimensionless . The radian is used in virtually all mathematical work beyond simple practical geometry , due , for example , to the pleasing and `` natural '' properties that the trigonometric functions display when their arguments are in radians . The radian is the ( derived ) unit of angular measurement in the SI system . Clock position ( n = 12 ) A clock position is the relative direction of an object described using the analogy of a 12 - hour clock . One imagines a clock face lying either upright or flat in front of oneself , and identifies the twelve hour markings with the directions in which they point . Hour angle ( n = 24 ) The astronomical hour angle is 1 / 24 of a turn . As this system is amenable to measuring objects that cycle once per day ( such as the relative position of stars ) , the sexagesimal subunits are called minute of time and second of time . These are distinct from , and 15 times larger than , minutes and seconds of arc . 1 hour = 15 ° = π / 12 rad = 1 / 6 quad. = 1 / 24 turn = 16 2 / 3 grad . ( Compass ) point or wind ( n = 32 ) The point , used in navigation , is 1 / 32 of a turn . 1 point = 1 / 8 of a right angle = 11.25 ° = 12.5 grad . Each point is subdivided in four quarter - points so that 1 turn equals 128 quarter - points . Hexacontade ( n = 60 ) The hexacontade is a unit of 6 ° that Eratosthenes used , so that a whole turn was divided into 60 units . Pechus ( n = 144 -- 180 ) -- The pechus was a Babylonian unit equal to about 2 ° or 2 1 / 2 ° . Binary degree ( n = 256 ) The binary degree , also known as the binary radian ( or brad ) , is 1 / 256 of a turn . The binary degree is used in computing so that an angle can be efficiently represented in a single byte ( albeit to limited precision ) . Other measures of angle used in computing may be based on dividing one whole turn into 2 equal parts for other values of n . Degree ( n = 360 ) The degree , denoted by a small superscript circle ( ° ) , is 1 / 360 of a turn , so one turn is 360 ° . The case of degrees for the formula given earlier , a degree of n = 360 ° units is obtained by setting k = 360 ° / 2π . One advantage of this old sexagesimal subunit is that many angles common in simple geometry are measured as a whole number of degrees . Fractions of a degree may be written in normal decimal notation ( e.g. 3.5 ° for three and a half degrees ) , but the `` minute '' and `` second '' sexagesimal subunits of the `` degree - minute - second '' system are also in use , especially for geographical coordinates and in astronomy and ballistics . Diameter part ( n = 376.99 ... ) The diameter part ( occasionally used in Islamic mathematics ) is 1 / 60 radian . One `` diameter part '' is approximately 0.95493 ° . There are about 376.991 diameter parts per turn . Grad ( n = 400 ) The grad , also called grade , gradian , or gon , is 1 / 400 of a turn , so a right angle is 100 grads . It is a decimal subunit of the quadrant . A kilometre was historically defined as a centi - grad of arc along a great circle of the Earth , so the kilometer is the decimal analog to the sexagesimal nautical mile . The grad is used mostly in triangulation . Milliradian The milliradian ( mil or mrad ) is defined as a thousandth of a radian , which means that a rotation of one turn consists of 2000π mil ( or approximately 6283.185 ... mil ) , and almost all scope sights for firearms are calibrated to this definition . In addition there are three other derived definitions used for artillery and navigation which are approximately equal to a milliradian . Under these three other definitions one turn makes up for exactly 6000 , 6300 or 6400 mils , which equals spanning the range from 0.05625 to 0.06 degrees ( 3.375 to 3.6 minutes ) . In comparison , the true milliradian is approximately 0.05729578 ... degrees ( 3.43775 ... minutes ) . One `` NATO mil '' is defined as 1 / 6400 of a circle . Just like with the true milliradian , each of the other definitions exploits the mil 's handby property of subtensions , i.e. that the value of one milliradian approximately equals the angle subtended by a width of 1 meter as seen from 1 km away ( 2π / 6400 = 0.0009817 ... ≈ 1 / 1000 ) . Minute of arc ( n = 21,600 ) The minute of arc ( or MOA , arcminute , or just minute ) is 1 / 60 of a degree = 1 / 21,600 turn . It is denoted by a single prime ( ′ ) . For example , 3 ° 30 ′ is equal to 3 × 60 + 30 = 210 minutes or 3 + 30 / 60 = 3.5 degrees . A mixed format with decimal fractions is also sometimes used , e.g. 3 ° 5.72 ′ = 3 + 5.72 / 60 degrees . A nautical mile was historically defined as a minute of arc along a great circle of the Earth . Second of arc ( n = 1,296,000 ) The second of arc ( or arcsecond , or just second ) is 1 / 60 of a minute of arc and 1 / 3600 of a degree . It is denoted by a double prime ( '' ) . For example , 3 ° 7 ′ 30 '' is equal to 3 + 7 / 60 + 30 / 3600 degrees , or 3.125 degrees . Positive and negative angles ( edit ) Although the definition of the measurement of an angle does not support the concept of a negative angle , it is frequently useful to impose a convention that allows positive and negative angular values to represent orientations and / or rotations in opposite directions relative to some reference . In a two - dimensional Cartesian coordinate system , an angle is typically defined by its two sides , with its vertex at the origin . The initial side is on the positive x-axis , while the other side or terminal side is defined by the measure from the initial side in radians , degrees , or turns . With positive angles representing rotations toward the positive y - axis and negative angles representing rotations toward the negative y - axis . When Cartesian coordinates are represented by standard position , defined by the x-axis rightward and the y - axis upward , positive rotations are anticlockwise and negative rotations are clockwise . In many contexts , an angle of − θ is effectively equivalent to an angle of `` one full turn minus θ '' . For example , an orientation represented as − 45 ° is effectively equivalent to an orientation represented as 360 ° − 45 ° or 315 ° . Although the final position is the same , a physical rotation ( movement ) of − 45 ° is not the same as a rotation of 315 ° ( for example , the rotation of a person holding a broom resting on a dusty floor would leave visually different traces of swept regions on the floor ) . In three - dimensional geometry , `` clockwise '' and `` anticlockwise '' have no absolute meaning , so the direction of positive and negative angles must be defined relative to some reference , which is typically a vector passing through the angle 's vertex and perpendicular to the plane in which the rays of the angle lie . In navigation , bearings are measured relative to north . By convention , viewed from above , bearing angles are positive clockwise , so a bearing of 45 ° corresponds to a north - east orientation . Negative bearings are not used in navigation , so a north - west orientation corresponds to a bearing of 315 ° . Alternative ways of measuring the size of an angle ( edit ) There are several alternatives to measuring the size of an angle by the angle of rotation . The grade of a slope , or gradient is equal to the tangent of the angle , or sometimes ( rarely ) the sine . A gradient is often expressed as a percentage . For very small values ( less than 5 % ) , the grade of a slope is approximately the measure of the angle in radians . In rational geometry the spread between two lines is defined as the square of the sine of the angle between the lines . As the sine of an angle and the sine of its supplementary angle are the same , any angle of rotation that maps one of the lines into the other leads to the same value for the spread between the lines . Astronomical approximations ( edit ) Astronomers measure angular separation of objects in degrees from their point of observation . 0.5 ° is approximately the width of the sun or moon . 1 ° is approximately the width of a little finger at arm 's length . 10 ° is approximately the width of a closed fist at arm 's length . 20 ° is approximately the width of a handspan at arm 's length . These measurements clearly depend on the individual subject , and the above should be treated as rough rule of thumb approximations only . Angles between curves ( edit ) The angle between the two curves at P is defined as the angle between the tangents A and B at P . The angle between a line and a curve ( mixed angle ) or between two intersecting curves ( curvilinear angle ) is defined to be the angle between the tangents at the point of intersection . Various names ( now rarely , if ever , used ) have been given to particular cases : -- amphicyrtic ( Gr . ἀμφί , on both sides , κυρτός , convex ) or cissoidal ( Gr . κισσός , ivy ) , biconvex ; xystroidal or sistroidal ( Gr . ξυστρίς , a tool for scraping ) , concavo - convex ; amphicoelic ( Gr . κοίλη , a hollow ) or angulus lunularis , biconcave . Bisecting and trisecting angles ( edit ) Main articles : Bisection § Angle bisector , and Angle trisection The ancient Greek mathematicians knew how to bisect an angle ( divide it into two angles of equal measure ) using only a compass and straightedge , but could only trisect certain angles . In 1837 Pierre Wantzel showed that for most angles this construction can not be performed . Dot product and generalisations ( edit ) In the Euclidean space , the angle θ between two Euclidean vectors u and v is related to their dot product and their lengths by the formula u ⋅ v = cos  ( θ ) ‖ u ‖ ‖ v ‖ . ( \ displaystyle \ mathbf ( u ) \ cdot \ mathbf ( v ) = \ cos ( \ theta ) \ left \ \ mathbf ( u ) \ right \ \ left \ \ mathbf ( v ) \ right \ . ) This formula supplies an easy method to find the angle between two planes ( or curved surfaces ) from their normal vectors and between skew lines from their vector equations . Inner product ( edit ) To define angles in an abstract real inner product space , we replace the Euclidean dot product ( ) by the inner product ⟨ ⋅ , ⋅ ⟩ ( \ displaystyle \ langle \ cdot , \ cdot \ rangle ) , i.e. ⟨ u , v ⟩ = cos  ( θ ) ‖ u ‖ ‖ v ‖ . ( \ displaystyle \ langle \ mathbf ( u ) , \ mathbf ( v ) \ rangle = \ cos ( \ theta ) \ \ left \ \ mathbf ( u ) \ right \ \ \ left \ \ mathbf ( v ) \ right \ . ) In a complex inner product space , the expression for the cosine above may give non-real values , so it is replaced with Re  ( ⟨ u , v ⟩ ) = cos  ( θ ) ‖ u ‖ ‖ v ‖ . ( \ displaystyle \ operatorname ( Re ) \ left ( \ langle \ mathbf ( u ) , \ mathbf ( v ) \ rangle \ right ) = \ cos ( \ theta ) \ \ left \ \ mathbf ( u ) \ right \ \ left \ \ mathbf ( v ) \ right \ . ) or , more commonly , using the absolute value , with ⟨ u , v ⟩ = cos  ( θ ) ‖ u ‖ ‖ v ‖ . ( \ displaystyle \ left \ langle \ mathbf ( u ) , \ mathbf ( v ) \ rangle \ right = \ cos ( \ theta ) \ \ left \ \ mathbf ( u ) \ right \ \ \ left \ \ mathbf ( v ) \ right \ . ) The latter definition ignores the direction of the vectors and thus describes the angle between one - dimensional subspaces span  ( u ) ( \ displaystyle \ operatorname ( span ) ( \ mathbf ( u ) ) ) and span  ( v ) ( \ displaystyle \ operatorname ( span ) ( \ mathbf ( v ) ) ) spanned by the vectors u ( \ displaystyle \ mathbf ( u ) ) and v ( \ displaystyle \ mathbf ( v ) ) correspondingly . Angles between subspaces ( edit ) The definition of the angle between one - dimensional subspaces span  ( u ) ( \ displaystyle \ operatorname ( span ) ( \ mathbf ( u ) ) ) and span  ( v ) ( \ displaystyle \ operatorname ( span ) ( \ mathbf ( v ) ) ) given by ⟨ u , v ⟩ = cos  ( θ ) ‖ u ‖ ‖ v ‖ ( \ displaystyle \ left \ langle \ mathbf ( u ) , \ mathbf ( v ) \ rangle \ right = \ cos ( \ theta ) \ left \ \ mathbf ( u ) \ right \ \ \ left \ \ mathbf ( v ) \ right \ ) in a Hilbert space can be extended to subspaces of any finite dimensions . Given two subspaces U ( \ displaystyle ( \ mathcal ( U ) ) ) , W ( \ displaystyle ( \ mathcal ( W ) ) ) with dim  ( U ) : = k ≤ dim  ( W ) : = l ( \ displaystyle \ dim ( ( \ mathcal ( U ) ) ) : = k \ leq \ dim ( ( \ mathcal ( W ) ) ) : = l ) , this leads to a definition of k ( \ displaystyle k ) angles called canonical or principal angles between subspaces . Angles in Riemannian geometry ( edit ) In Riemannian geometry , the metric tensor is used to define the angle between two tangents . Where U and V are tangent vectors and g are the components of the metric tensor G , cos  θ = g i j U i V j g i j U i U j g i j V i V j . ( \ displaystyle \ cos \ theta = ( \ frac ( g_ ( ij ) U ^ ( i ) V ^ ( j ) ) ( \ sqrt ( \ left g_ ( ij ) U ^ ( i ) U ^ ( j ) \ right \ left g_ ( ij ) V ^ ( i ) V ^ ( j ) \ right ) ) ) . ) Hyperbolic angle ( edit ) A hyperbolic angle is an argument of a hyperbolic function just as the circular angle is the argument of a circular function . The comparison can be visualized as the size of the openings of a hyperbolic sector and a circular sector since the areas of these sectors correspond to the angle magnitudes in each case . Unlike the circular angle , the hyperbolic angle is unbounded . When the circular and hyperbolic functions are viewed as infinite series in their angle argument , the circular ones are just alternating series forms of the hyperbolic functions . This weaving of the two types of angle and function was explained by Leonhard Euler in Introduction to the Analysis of the Infinite . Angles in geography and astronomy ( edit ) In geography , the location of any point on the Earth can be identified using a geographic coordinate system . This system specifies the latitude and longitude of any location in terms of angles subtended at the centre of the Earth , using the equator and ( usually ) the Greenwich meridian as references . In astronomy , a given point on the celestial sphere ( that is , the apparent position of an astronomical object ) can be identified using any of several astronomical coordinate systems , where the references vary according to the particular system . Astronomers measure the angular separation of two stars by imagining two lines through the centre of the Earth , each intersecting one of the stars . The angle between those lines can be measured , and is the angular separation between the two stars . In both geography and astronomy , a sighting direction can be specified in terms of a vertical angle such as altitude / elevation with respect to the horizon as well as the azimuth with respect to north . Astronomers also measure the apparent size of objects as an angular diameter . For example , the full moon has an angular diameter of approximately 0.5 ° , when viewed from Earth . One could say , `` The Moon 's diameter subtends an angle of half a degree . '' The small - angle formula can be used to convert such an angular measurement into a distance / size ratio . See also ( edit ) Angle bisector Angular velocity Argument ( complex analysis ) Astrological aspect Central angle Clock angle problem Dihedral angle Exterior angle theorem Great circle distance Inscribed angle Irrational angle Phase angle Protractor Solid angle for a concept of angle in three dimensions . Spherical angle Transcendent angle Trisection Zenith angle Notes ( edit ) Jump up ^ This approach requires however an additional proof that the measure of the angle does not change with changing radius r , in addition to the issue of `` measurement units chosen . '' A smoother approach is to measure the angle by the length of the corresponding unit circle arc . Here `` unit '' can be chosen to be dimensionless in the sense that it is the real number 1 associated with the unit segment on the real line . See R. Dimitric for instance . Jump up ^ Sidorov 2001 Jump up ^ Slocum 2007 Jump up ^ Chisholm 1911 ; Heiberg 1908 , pp. 177 -- 178 Jump up ^ `` Mathwords : Reference Angle '' . www.mathwords.com . Archived from the original on 23 October 2017 . Retrieved 26 April 2018 . Jump up ^ Wong & Wong 2009 , pp. 161 -- 163 Jump up ^ Euclid ( c. 300 BC ) . The Elements . Check date values in : date = ( help ) Proposition I : 13 . ^ Jump up to : Shute , Shirk & Porter 1960 , pp. 25 -- 27 . Jump up ^ Jacobs 1974 , p. 255 . ^ Jump up to : Chisholm 1911 Jump up ^ Jacobs 1974 , p. 97 . Jump up ^ Henderson & Taimina 2005 , p. 104 . ^ Jump up to : Johnson , Roger A. Advanced Euclidean Geometry , Dover Publications , 2007 . Jump up ^ D. Zwillinger , ed. ( 1995 ) , CRC Standard Mathematical Tables and Formulae , Boca Raton , FL : CRC Press , p. 270 as cited in Weisstein , Eric W. `` Exterior Angle '' . MathWorld . Jump up ^ R. Dimitric : On angles and angle measurements Archived 2013 - 09 - 27 at the Wayback Machine . Jump up ^ J.H. Jeans ( 1947 ) , The Growth of Physical Science , p. 7 ; Francis Dominic Murnaghan ( 1946 ) , Analytic Geometry , p. 2 Jump up ^ ooPIC Programmer 's Guide ( archived ) oopic.com Jump up ^ Angles , integers , and modulo arithmetic Archived 2011 - 06 - 30 at Wikiwix Shawn Hargreaves blogs.msdn.com Jump up ^ Chisholm 1911 ; Heiberg 1908 , p. 178 References ( edit ) Henderson , David W. ; Taimina , Daina ( 2005 ) , Experiencing Geometry / Euclidean and Non-Euclidean with History ( 3rd ed . ) , Pearson Prentice Hall , p. 104 , ISBN 9780131437487 Heiberg , Johan Ludvig ( 1908 ) , Heath , T.L. , ed. , Euclid , The Thirteen Books of Euclid 's Elements , 1 , Cambridge : Cambridge University Press . Sidorov , L.A. ( 2001 ) ( 1994 ) , `` Angle '' , in Hazewinkel , Michiel , Encyclopedia of Mathematics , Springer Science + Business Media B.V. / Kluwer Academic Publishers , ISBN 978 - 1 - 55608 - 010 - 4 Jacobs , Harold R. ( 1974 ) , Geometry , W.H. Freeman , pp. 97 , 255 , ISBN 0 - 7167 - 0456 - 0 Slocum , Jonathan ( 2007 ) , Preliminary Indo - European lexicon -- Pokorny PIE data , University of Texas research department : linguistics research center , retrieved 2 Feb 2010 Shute , William G. ; Shirk , William W. ; Porter , George F. ( 1960 ) , Plane and Solid Geometry , American Book Company , pp. 25 -- 27 Wong , Tak - wah ; Wong , Ming - sim ( 2009 ) , `` Angles in Intersecting and Parallel Lines '' , New Century Mathematics , 1B ( 1 ed . ) , Hong Kong : Oxford University Press , pp. 161 -- 163 , ISBN 978 - 0 - 19 - 800177 - 5 Attribution This article incorporates text from a publication now in the public domain : Chisholm , Hugh , ed. ( 1911 ) , `` Angle '' , Encyclopædia Britannica , 2 ( 11th ed . ) , Cambridge University Press , p. 14 External links ( edit ) Wikimedia Commons has media related to Angles ( geometry ) . `` Angle '' , Encyclopædia Britannica , 2 ( 9th ed . ) , 1878 , pp. 29 -- 30 Proximity construction of an angle in decimal degrees with the third intercept theorem Angle Bisectors in a Quadrilateral at cut - the - knot Constructing a triangle from its angle bisectors at cut - the - knot Various angle constructions with compass and straightedge Complementary Angles animated demonstration . With interactive applet Supplementary Angles animated demonstration . With interactive applet Angle definition pages with interactive applets that are also useful in a classroom setting . Math Open Reference Construction of an angle Site geometry GND : 4189964 - 7 Retrieved from `` https://en.wikipedia.org/w/index.php?title=Angle&oldid=838407073 '' Categories : Angle Hidden categories : CS1 errors : dates Webarchive template wayback links Webarchive template other archives Articles containing Ancient Greek - language text Pages using div col with deprecated parameters Wikipedia articles incorporating a citation from the 1911 Encyclopaedia Britannica with Wikisource reference Wikipedia articles incorporating text from the 1911 Encyclopædia Britannica Commons category with local link different than on Wikidata Wikipedia articles incorporating a citation from EB9 All articles with dead external links Articles with dead external links from October 2016 Articles with permanently dead external links Wikipedia articles with GND identifiers Talk Contents About Wikipedia Адыгэбзэ Afrikaans Alemannisch Aragonés ܐܪܡܝܐ Asturianu Avañe'ẽ Aymar aru Azərbaycanca تۆرکجه বাংলা Башҡортса Беларуская Беларуская ( тарашкевіца )  Български Bosanski Brezhoneg Буряад Català Čeština ChiShona Cymraeg Dansk Deutsch Eesti Ελληνικά Español Esperanto Euskara فارسی Français Gaeilge Gàidhlig Galego 贛 語 한국어 Հայերեն हिन्दी Hrvatski Ido Ilokano Bahasa Indonesia Interlingua Italiano עברית ქართული Қазақша Kernowek Kiswahili Kreyòl ayisyen Kurdî Latina Latviešu Lietuvių Limburgs Lingála Magyar Македонски മലയാളം मराठी Bahasa Melayu Монгол Nederlands नेपाली नेपाल भाषा 日本 語 Nordfriisk Norsk Norsk nynorsk Occitan Олык марий Oʻzbekcha / ўзбекча ਪੰਜਾਬੀ پنجابی Patois ភាសា ខ្មែរ Polski Português Română Runa Simi Русский Scots Sicilianu සිංහල Simple English Slovenčina Slovenščina Soomaaliga کوردی Српски / srpski Srpskohrvatski / српскохрватски Basa Sunda Suomi Svenska Tagalog தமிழ் Татарча / tatarça ไทย ᏣᎳᎩ Türkçe Українська اردو Tiếng Việt 文言 Winaray ייִדיש 粵語 中文 100 more Edit links This page was last edited on 26 April 2018 , at 20 : 15 . 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Question: points on a sphere or angles in a circle are measured in units called


Climate of Dubai - wikipedia Climate of Dubai Jump to : navigation , search Clouds over Dubai in winter . Dubai has a tropical desert climate , Köppen classification Bwh , because of its location within the Northern desert belt . Summers are extremely hot and humid , with an average high around 41 ° C ( 106 ° F ) and overnight lows around 30 ° C ( 86 ° F ) . Most days are sunny throughout the year . Winters are warm and short with an average high of 23 ° C ( 73 ° F ) and overnight lows of 14 ° C ( 57 ° F ) . Precipitation , however , has been increasing in the last few decades with accumulated rain reaching 150 mm ( 5.91 in ) per year . The weather in Dubai can bring short and irregular rainfall as is typical for the Middle East . Most of the rainfall occurs in the December to March period . The weather between December and March remains warm and is considered to be the most comfortable climatic conditions of the year . Contents ( hide ) 1 Climate data 2 Seasonal climate 3 Climatic conditions by month 4 Sandstorms 5 References 6 External links Climate data ( edit ) ( hide ) Climate data for Dubai Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Year Record high ° C ( ° F ) 31.8 ( 89.2 ) 37.5 ( 99.5 ) 41.3 ( 106.3 ) 43.5 ( 110.3 ) 47.0 ( 116.6 ) 47.9 ( 118.2 ) 48.5 ( 119.3 ) 47.5 ( 117.5 ) 45.1 ( 113.2 ) 42.4 ( 108.3 ) 38 ( 100 ) 33.2 ( 91.8 ) 48.5 ( 119.3 ) Average high ° C ( ° F ) 16.2 ( 61.2 ) 16.3 ( 61.3 ) 18.6 ( 65.5 ) 22.2 ( 72 ) 27.0 ( 80.6 ) 32.7 ( 90.9 ) 35.2 ( 95.4 ) 36.4 ( 97.5 ) 33.1 ( 91.6 ) 24.6 ( 76.3 ) 19.7 ( 67.5 ) 17.3 ( 63.1 ) 25.4 ( 77.7 ) Daily mean ° C ( ° F ) 15.3 ( 59.5 ) 15.6 ( 60.1 ) 17.2 ( 63 ) 20.2 ( 68.4 ) 25.4 ( 77.7 ) 28.6 ( 83.5 ) 30.7 ( 87.3 ) 30.0 ( 86 ) 28.4 ( 83.1 ) 22.0 ( 71.6 ) 17.4 ( 63.3 ) 15 . ( 59 ) 22.15 ( 71.88 ) Average low ° C ( ° F ) 14.4 ( 57.9 ) 15.5 ( 59.9 ) 17.7 ( 63.9 ) 21.2 ( 70.2 ) 24.9 ( 76.8 ) 27.5 ( 81.5 ) 28.2 ( 82.8 ) 28.5 ( 83.3 ) 25.7 ( 78.3 ) 24.3 ( 75.7 ) 20.0 ( 68 ) 16.3 ( 61.3 ) 22.5 ( 72.5 ) Record low ° C ( ° F ) 6.1 ( 43 ) 6.9 ( 44.4 ) 9.0 ( 48.2 ) 13.4 ( 56.1 ) 15.1 ( 59.2 ) 18.2 ( 64.8 ) 20.4 ( 68.7 ) 23.1 ( 73.6 ) 16.5 ( 61.7 ) 15.0 ( 59 ) 11.8 ( 53.2 ) 8.2 ( 46.8 ) 6.1 ( 43 ) Average precipitation mm ( inches ) 18.8 ( 0.74 ) 25.0 ( 0.984 ) 22.1 ( 0.87 ) 7.2 ( 0.283 ) 0.4 ( 0.016 ) 0.0 ( 0 ) 0.8 ( 0.031 ) 0.0 ( 0 ) 0.0 ( 0 ) ( 0.043 ) 2.7 ( 0.106 ) 16.2 ( 0.638 ) 94.3 ( 3.711 ) Average precipitation days 5.4 4.7 5.8 2.6 0.3 0.0 0.5 0.5 0.1 0.2 1.3 3.8 25.2 Average relative humidity ( % ) 65 65 63 55 53 58 56 57 60 60 61 64 59.8 Mean monthly sunshine hours 254.2 229.6 254.2 294.0 344.1 342.0 322.4 316.2 309.0 303.8 285.0 254.2 3,508.7 Percent possible sunshine 75 75 68 75 85 81 74 78 86 82 86 75 78.3 Source # 1 : Dubai Meteorological Office Source # 2 : climatebase.ru ( extremes , sun ) , NOAA ( humidity , 1974 -- 1991 ) Climate data for Dubai Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Year Average sea temperature ° C ( ° F ) 23.4 ( 74.2 ) 21.9 ( 71.4 ) 23.2 ( 73.8 ) 25.5 ( 77.9 ) 28.8 ( 83.8 ) 31.6 ( 88.8 ) 32.7 ( 90.9 ) 33.5 ( 92.3 ) 33.1 ( 91.5 ) 31.3 ( 88.4 ) 28.6 ( 83.4 ) 25.4 ( 77.8 ) 28.3 ( 76.6 ) Average Ultraviolet index 6 8 10 11 + 11 + 11 + 11 + 11 + 11 8 6 5 9.1 Source # 1 : seatemperature.org Source # 2 : Weather Atlas Seasonal climate ( edit ) The climate of Dubai is warm and sunny due to its position near the line of the Tropic of Cancer . During the winter season it has an average daytime temperature of 25 ° C ( 77 ° F ) . Nighttime temperatures near the coastline range between 12 ° C ( 54 ° F ) - 15 ° C ( 59 ° F ) , while in the desert they are 5 ° C ( 41 ° F ) with the nights being relatively cool throughout the year . Near coastal areas humidity averages between 50 % and 60 % . In the summer , the weather in Dubai is very hot , dry and humid , with temperatures reaching 45 ° C ( 113 ° F ) for many days . Even the sea temperature reaches 37 ° C ( 99 ° F ) , with humidity averaging over 90 % . Rainfall in Dubai is infrequent and does not last for a long period . It mostly rains during the winter period between December and March in the form of short downpours and an occasional thunderstorm . On average , rain falls only five days a year . February is the wettest month in Dubai with an average of 35 millimetres ( 1.4 in ) of rain . The weather in Dubai is extremely dry in the month of June with little or no rain . March and December also record some amount of rainfall . The rainfall during January , April , July , October and November are about average while the amount of rain in May , August and September are comparatively lower . Climatic conditions by month ( edit ) In January the average maximum daytime temperature in Dubai is 24 ° C ( 75 ° F ) . The average rainfall is 10 mm ( 0.4 inches ) of , with rain generally falling on just two days of the month . While the average sea temperature in January is 22 ° C ( 72 ° F ) . On January 15 , 2008 110 millimetres ( 4.3 in ) of rain was recorded in 24 hours . In February the average maximum daytime temperature in Dubai is 25 ° C ( 77 ° F ) . The average rainfall is 35 millimetres ( 1.4 in ) during February . While the average sea temperature is 22 ° C ( 72 ° F ) . In March the average temperature is 28 ° C ( 82 ° F ) , with moderate heat and humidity . The average rainfall is 27 millimetres ( 1.1 in ) during March , with rain generally falling on just four days of the month . While the average sea temperature is 23 ° C ( 73 ° F ) . In April the average daytime temperature is 32 ° C ( 90 ° F ) , with high heat and humidity . The average rainfall is 8 millimetres ( 0.31 in ) with rain generally falling on just two days of the month . The average sea temperature in April is 24 ° C ( 75 ° F ) . In May the average temperature is 38 ° C ( 100 ° F ) , with extreme heat and humidity . The average rainfall is 1 millimetre ( 0.039 in ) during May , with rain generally falling on just one day of the month . The average sea temperature is 28 ° C ( 82 ° F ) . In June the average temperature is 40 ° C ( 104 ° F ) , with extreme heat and humidity . There is very little chance of any rainfall in June . The average sea temperature in June is 31 ° C ( 88 ° F ) . In July the average temperature is extremely high 42 ° C ( 108 ° F ) with extreme heat and humidity . Average sea temperature in the month of July is 32 ° C ( 90 ° F ) . In August the average temperature is 43 ° C ( 109 ° F ) , with extreme heat and humidity . There is very little chance of any rainfall during August . While the average sea temperature in August is 32 ° C ( 90 ° F ) . In September the average temperature is 39 ° C ( 102 ° F ) , with extreme heat and humidity . There is very little chance of any rainfall during September . Average sea temperature in September is 32 ° C ( 90 ° F ) . In October the average temperature is 36 ° C ( 97 ° F ) , with very high heat and humidity . The average sea temperature around Dubai Airport in October is 30 ° C ( 86 ° F ) . In November the average temperature is 31 ° C ( 88 ° F ) , with high heat and humidity . On average only 3 millimetres ( 0.12 in ) of rainfall may occur in November . Average sea temperature in November is 28 ° C ( 82 ° F ) . In December the average temperature is 26 ° C ( 79 ° F ) , cool to warm with high humidity . The average rainfall during the month is 15 millimetres ( 0.59 in ) . With rain generally falling on just 2 days of the month . The average sea temperature in December is 25 ° C ( 77 ° F ) . Sandstorms ( edit ) This section needs additional citations for verification . Please help improve this article by adding citations to reliable sources . Unsourced material may be challenged and removed . ( April 2018 ) ( Learn how and when to remove this template message ) During the summer season , a low pressure area develops over Dubai forcing strong north - westerly winds to blow from Saudi Arabia . These winds , also known as Shamal ( north ) in Arabic , become gusty and unpredictable on reaching Dubai . Shamal boost up the desert sand and reduce visibility and the sandstorms may last for several days. Following are some notable sandstorms that hit Dubai during the last few years . On April 18 , 2008 a severe sandstorm hits Dubai and reduced visibility to 1,500 meters . On March 10 , 2009 a moderate sandstorm reached Dubai which lowered down the visibility and temperature . On March 2 , 2010 Strong winds caused sandstorms in Dubai and other regions of United Arab Emirates . On January 28 , 2011 a sandstorm hit Dubai for a short time . On February 26 , 2012 a strong sandstorm hit Dubai and other parts of the country On April 2 , 2015 a very strong sandstorm hit Dubai and other parts of the country . It started to clear out closer to the evening . It reduced the visibility to 3000 metres . References ( edit ) Jump up ^ `` Dubai Humidity and Temperatures : Average Temperatures and Humidity of Dubai '' . holiday-weather.com . Retrieved 2014 - 10 - 19 . Jump up ^ Climate in Dubai across the year Archived December 18 , 2010 , at the Wayback Machine ... Dubai Meteorological office . Jump up ^ `` Climate ( Average Temperatures : 1977 -- 2015 ; Precipitation : 1967 - 2009 ) '' . Dubai Meteorological Office . Archived from the original on 4 October 2013 . Retrieved 20 December 2008 . CS1 maint : BOT : original - url status unknown ( link ) Jump up ^ `` Dubai , Emirates '' . Climatebase.ru . Retrieved 10 February 2013 . Jump up ^ `` Climate Normals for Dubai '' . National Oceanic and Atmospheric Administration . Retrieved 10 February 2013 . Jump up ^ `` Monthly Dubai water temperature chart '' . Seatemperatures.org . Retrieved 20 January 2014 . Jump up ^ `` Dubai , United Arab Emirates - Climate data '' . Weather Atlas . Retrieved 8 March 2017 . ^ Jump up to : `` Dubai Geography - Information , climate and weather in Dubai '' . Dubai.com. 2013 - 05 - 27 . Retrieved 2013 - 06 - 01 . ^ Jump up to : `` Weather in Dubai City - Information on the Weather in Dubai , UAE '' . Visit-dubai-city.com . Retrieved 2013 - 06 - 01 . ^ Jump up to : Weather2Flights.com . `` Airports - Who flies there ? '' . Weather2flights.com . Retrieved 2013 - 06 - 01 . Jump up ^ `` Downpour continues as Dubai receives record rainfall '' . GulfNews.com. 2008 - 01 - 16 . Retrieved 2013 - 06 - 01 . Jump up ^ `` Severe sandstorm reduces visibility in Dubai '' . GulfNews.com. 2008 - 04 - 19 . Retrieved 2013 - 06 - 01 . Jump up ^ `` Moderate sandstorm reaches Dubai '' . GulfNews.com. 2009 - 03 - 11 . Retrieved 2013 - 06 - 01 . External links ( edit ) Dubai weather - UAE 's National Center of Meteorology and Seismology Weather of Dubai Dreamsofdubai Climate-Charts.com Dubai Climate Change Adaptation Plan - Emirates 24 / 7 Dubai topics History Timeline Geography Government Culture History Trucial Oman Al Maktoum dynasty Bani Yas Geography Deira Bur Dubai Dubai Creek Deira Corniche Government Mohammed bin Rashid Al Maktoum Dubai Police Force Dubai Municipality Education American University in Dubai Dubai Men 's College Dubai Women 's College Birla Institute of Technology and Science Zayed University SP Jain Center of Management Mahatma Gandhi University Our Own English High School The Indian High School Dubai Modern High School Dubai British School English College Dubai International School of Choueifat American School of Dubai Dubai American Academy Religion Grand Mosque Hindu Temple St. Mary 's Catholic Church St. Francis of Assisi Catholic Church Demographics Emiratis Expatriates Indians Pakistanis List of people from Dubai Towns Buildings Places Structures Towns Dubai Hatta Al Aweer Al Shindagha Buildings and structures List of buildings in Dubai 21st Century Tower Burj Al Arab Burj Khalifa Chelsea Tower Dubai Frame Dubai World Trade Centre Emirates Towers Etisalat Tower 1 Etisalat Tower 2 more Parks Mushrif Park Safa Park Zabeel Park Real estate Palm Islands The World Dubai Marina Dubai Waterfront Business Bay Dubailand Bawadi Dubai South Dubai Festival City Jumeirah Beach Residence Palm Jumeirah Palm Jebel Ali Palm Deira Bridges and tunnels Al Shindagha Tunnel Al Maktoum Bridge Floating Bridge Al Garhoud Bridge Business Bay Crossing Sheikh Rashid bin Saeed Crossing Economy Trade Commerce Trade and commerce Emaar Nakheel Dubai Holding Al - Ghurair Group Ashai Group International Etisalat Du Dubai Ports World Art Dubai Gold Souk / Spice Souk Ports Port Rashid Jebel Ali Al Hamriya Port Port Saeed Free zones Dubai International Financial Centre Dubai Internet City Dubai Knowledge Village Dubai Media City Dubai Production City Dubai Silicon Oasis Dubai Studio City Jebel Ali Shopping malls Mall of the Emirates BurJuman Ibn Battuta Mall Dubai Mall Deira City Centre Transport Transport Al Maktoum International Airport Dubai International Airport Abra Dubai Metro Red Line Green Line Dubai Tram Dubai Trolley Dubai Water Canal Emirates Airlines Palm Jumeirah Monorail Routes E 11 E 311 E 44 E 66 E 77 E 611 D 94 D 92 D 89 D 85 D 75 D 73 D 90 Sister cities Chennai Detroit Geneva Shanghai Casablanca Frankfurt Guangzhou Osaka Beirut Dundee Gold Coast Moscow Istanbul Damascus Barcelona Portal WikiProject Retrieved from `` https://en.wikipedia.org/w/index.php?title=Climate_of_Dubai&oldid=838253302 '' Categories : Geography of Dubai Climate of the United Arab Emirates Climate by city Hidden categories : Webarchive template wayback links CS1 maint : BOT : original - url status unknown Articles needing additional references from April 2018 All articles needing additional references Talk Contents About Wikipedia فارسی Edit links This page was last edited on 25 April 2018 , at 21 : 37 . 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Question: what is the year round weather in dubai


John Hart ( actor ) - wikipedia John Hart ( actor ) Jump to : navigation , search This article is about the American cinema actor . For other people with the same name , see John Hart ( disambiguation ) . John Hart John Hart portrays Nat `` Hawkeye '' Cutler in the syndicated 1957 Western series `` Hawkeye and the Last of the Mohicans . '' ( 1917 - 12 - 13 ) December 13 , 1917 Los Angeles , California , U.S. September 20 , 2009 ( 2009 - 09 - 20 ) ( aged 91 ) Playas de Rosarito , Baja California , Mexico Cause of death Complications from dementia Resting place Cremated Nationality American Other names John Hilton Education South Pasadena High School Occupation Actor Years active 1937 -- 1982 Spouse ( s ) Beryl Braithwaite ( m . 1957 -- 2009 ) Children 2 ; including Buddy Hart John Hart ( December 13 , 1917 -- September 20 , 2009 ) , also credited as John Hilton was an American film and television actor . In his early career , Hart appeared mostly in Westerns . Although Hart played mostly minor roles in some fairly well known films , he was probably best known for having replaced Clayton Moore in 1952 in the television series , The Lone Ranger for one season from 1952 until 1953 . Contents ( hide ) 1 Career 2 Personal and Death 3 Filmography 4 References 5 External links Career ( edit ) Hart began his screen career in 1937 with a bit part in Daughter of Shanghai . He continued in a variety of B pictures such as Prison Farm and King of Alcatraz before appearing in two of Cecil B. DeMille 's films The Buccaneer ( 1938 ) and North West Mounted Police ( 1940 ) . In 1941 , Hart 's acting career was interrupted when he was drafted into the United States Army . He rose to the rank of first sergeant in the Coast Artillery and eventually served in the Philippines . Following his military service , Hart worked frequently for Sam Katzman ; he was given the lead role in the Jack Armstrong ( 1947 ) film serial . Hart did stunt work and acted in numerous Westerns Hart was eventually offered the opportunity to replace Clayton Moore on The Lone Ranger television series . Based on the assumption that the masked character , rather than the actor , was the true star of The Lone Ranger , the program 's producers fired Moore ( presumably over salary differences ) and replaced him with Hart , who was of a similar build and had a comparable background in Westerns . However , the public never truly accepted Hart as the Lone Ranger , and by 1954 the producers returned Moore to the role . According to Clayton Moore 's autobiography I Was That Masked Man , Moore never knew why he was replaced by Hart , and also stated that he had not sought a pay increase . Hart acted in minor roles in two episodes of The Lone Ranger before being asked to replace Clayton Moore for the entire third season . The episodes were `` Rifles and Renegades '' ( # 34 ) and `` Sheriff at Gunstock '' ( # 46 ) . Hart continued to act in films for more than two decades , appearing in films of several genres , almost always in supporting roles . Hart appeared twice in on the TV series I Love Lucy as Lucy 's old boyfriend and again in the Hollywood episodes as a lifeguard at the hotel pool . In 1955 , Hart starred in The Adventures of Captain Africa , which was originally intended to be a new movie about famous comic book hero The Phantom . However , licensing issues forced Columbia Pictures re-film the entire serial and re-christen the hero `` Captain Africa . '' Hart also had numerous supporting roles in the Highway Patrol TV series . In 1957 , Hart portrayed Nat `` Hawkeye '' Cutler in the syndicated western series Hawkeye and the Last of the Mohicans , . The one - season program was based loosely on the novels of James Fenimore Cooper . It was filmed in Canada and featured some Indians in a favorable image . Interviewed by Tom Weaver in `` Western Clippings '' magazine ( May -- June 2000 ) , Hart revealed that he met actress Beryl Braithwaite when she played a role in an episode of the series ; they married just days later , with `` Hawkeye '' co-star Lon Chaney , Jr. , the best man at the wedding . In 1965 Hart made two brief appearances on the TV series Perry Mason , including the role of title character and murder victim Jamison Selff in `` The Case of the Wrathful Wraith . '' In the 1970 film The Phynx , Hart played the Lone Ranger alongside Jay Silverheels as Tonto , spoofing their characters . Hart 's last theatrical film appearance was in 1981 's The Legend of the Lone Ranger in which he appeared as a newspaper editor . He appeared in the television series Happy Days as the Lone Ranger in the episode `` Hi Yo , Fonzie Away '' ( February 9 , 1982 ) . In this episode Fonzie , played by Henry Winkler gets to meet his childhood hero , the Lone Ranger , for his birthday . Hart 's other major late appearance was in an episode of The Greatest American Hero , My Heroes Have Always Been Cowboys , where Hart gives the title character an inspiring speech about heroism . Personal and death ( edit ) Born in Los Angeles , Hart 's mother was named Enid and he had a sister ; Shari . Hart attended South Pasadena High School and later studied acting at the Pasadena Playhouse . On September 20 , 2009 , Hart died from complications of dementia at his home in Playas de Rosarito , Baja California , Mexico at the age of 91 , survived by his wife Beryl , a son and a daughter . His ashes were scattered into the Pacific Ocean . Filmography ( edit ) 1937 : Daughter of Shanghai 1938 : The Buccaneer 1938 : King of Alcatraz 1938 : Prison Farm 1939 : Disbarred 1940 : North West Mounted Police 1947 : Vacation Days 1949 : Batman and Robin 1950 : Atom Man vs. Superman 1951 : Stagecoach Driver 1952 : Kansas Territory 1952 : Texas City 1953 : The Lone Ranger ( TV series ) 1953 : The Great Adventures of Captain Kidd 1953 : I Love Lucy ( 3 episodes , 1953 - 55 ) 1955 : The Adventures of Captain Africa ( serial ) 1956 : The Ten Commandments 1957 : Hawkeye and the Last of the Mohicans ( TV series ) 1958 : Leave It to Beaver ( 3 episodes , 1958 - 62 ) 1961 : Pete and Gladys as Mervin in `` The Fur Coat Story '' ( CBS - TV ) 1962 : The Horizontal Lieutenant 1965 : Rawhide ( CBS - TV ) 1966 : Django spara per primo 1970 : The Phynx 1973 : Santee 1973 : Blackenstein 1974 : Welcome to Arrow Beach 1976 : Blood Voyage 1976 : The Astral Factor 1978 : Cheerleaders Beach Party 1981 : The Greatest American Hero ( ABC - TV episode ) 1981 : The Legend of the Lone Ranger 1982 : Happy Days ( ABC - TV episode ) Biography portal Los Angeles portal California portal Mexico portal Film portal Television portal References ( edit ) Jump up ^ John Hart - NNDB Jump up ^ Giesea , Roberta ( 2000 ) . `` The Lone Ranger Comes to Rosarito '' . Baja California , Mexico : Baja Times . Retrieved September 8 , 2014 . Jump up ^ McLellan , Dennis ( June 12 , 1993 ) . `` After 60 Years , the Lone Ranger Still Lives '' . Los Angeles Times . Retrieved September 27 , 2010 . ^ Jump up to : McLellan , Dennis ( September 29 , 2009 ) . `` John Hart dies 91 '' . Los Angeles , California : Los Angeles Times . Retrieved September 8 , 2014 . Jump up ^ `` ( photo caption ) '' . Films of the Golden Age ( 81 ) : 53 . Summer 2015 . External links ( edit ) John Hart at Find a Grave John Hart on IMDb Hawkeye and the Last of the Mohicans Page VIAF : 26622371 ISNI : 0000 0000 4326 3960 Retrieved from `` https://en.wikipedia.org/w/index.php?title=John_Hart_(actor)&oldid=792024519 '' Categories : 1917 births 2009 deaths Male actors from Los Angeles American military personnel of World War II American male film actors American male television actors Western ( genre ) television actors Hidden categories : Pages to import images to Wikidata Articles with hCards Find a Grave template with ID same as Wikidata Wikipedia articles with VIAF identifiers Wikipedia articles with ISNI identifiers Talk Contents About Wikipedia Asturianu Deutsch Español فارسی Français Polski Edit links This page was last edited on 24 July 2017 , at 00 : 13 . Text is available under the Creative Commons Attribution - ShareAlike License ; additional terms may apply . By using this site , you agree to the Terms of Use and Privacy Policy . 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Question: when did john hart play the lone ranger


De pictura - wikipedia De pictura Jump to : navigation , search Figure from Della pittura showing the vanishing point Alberti 's figure showing how a circle projected as an ellipse Figure showing pillars in perspective on a grid De pictura ( English : `` On Painting '' ) is a treatise written by the Italian architect and art theorist Leon Battista Alberti . The first version , written in vernacular Italian in 1435 under the title Della pittura , was for a general audience . The Latin version , the De pictura of 1439 -- 41 , was more technical and intended for scholars . The work is the first in a trilogy of treatises on the `` Major arts '' which had a widespread circulation during the Renaissance , the others being De re aedificatoria ( `` On Architecture '' , 1454 ) and De statua ( `` On Sculpture '' , 1462 ) . Contents ( hide ) 1 Context 2 Work 2.1 Contents 2.2 Approach 3 Influence 4 Bibliography 4.1 Editions 4.2 Translations and critical editions 5 References Context ( edit ) Alberti was a member of Florentine family exiled in the 14th century , who was able to return in Florence only from 1434 , in the following of the Papal court during the Council of Florence . Here he knew contemporary art innovators such as Filippo Brunelleschi , Donatello and Masaccio , with whom he shared an interest for Renaissance humanism and classical art . Alberti was the first post-classical writer to produce a work of art theory , as opposed to works about the function of religious art or art techniques , and reflected the developing Italian Renaissance art of his day . Work ( edit ) Contents ( edit ) Book I : a simple introduction for young boys , in preparation for studying painting Book II : a survey of types of painting for teenage trainees in a workshop Book III : advice to the adult painter on how to perfect his skill Approach ( edit ) Further information : Mathematics and art De pictura aimed to describe systematically the figurative arts through `` geometry '' . Alberti divided painting into three parts : Circumscriptio ( Italian : disegno ) , consisting in drawing the bodies ' contour Compositio ( commensuratio in the Italian version of the treatise ) , including tracing the lines joining the bodies Receptio luminum ( color ) , taking into consideration colors and light . The treatise contained an analysis of all the techniques and painting theories known at the time , in this surpassing medieval works such as The book of Art by Cennino Cennini ( 1390 ) . De pictura also includes the first description of linear geometric perspective around 1416 ; Alberti credited the discovery to Brunelleschi , and dedicated the 1435 edition to him . Alberti argued that multi-figure history painting was the noblest form of art , as being the most difficult , which required mastery of all the others , because it was a visual form of history , and because it had the greatest potential to move the viewer . He placed emphasis on the ability to depict the interactions between the figures by gesture and expression . De pictura relied heavily on references to art in classical literature ; in fact Giotto 's huge Navicella in mosaic at Old St. Peter 's Basilica in Rome ( now effectively lost ) was the only modern ( post-classical ) work described in it . Influence ( edit ) The accurate perspective in Leonardo da Vinci 's paintings such as his Annunciation ( 1475 -- 1480 ) derives from Alberti 's De pictura . De pictura influenced the work of artists including Donatello , Ghiberti , Botticelli , and Ghirlandaio . His treatment of perspective was the most influential of his recommendations , having a powerful effect on Leonardo da Vinci , and through him on the whole Italian renaissance . Alberti made at least 29 uses of Pliny the Elder 's Natural History , deriving his key themes of simplicity and seriousness directly from Pliny . For example , Alberti advised artists to use colour with restraint , and to paint in the effect of gold rather than using actual gold in their paintings . Gold did indeed vanish from Italian paintings of the second part of the 15th century . Artists however found their own ways to paint with restraint , rather than following Alberti 's actual instructions directly . Similarly , he encouraged artists to add black when modelling shapes , rather than only adding white as Cennino Cennini had advised in his c. 1390 Il Libro dell'Arte . This advice had the effect of making Italian renaissance paintings more sombre . Alberti was here perhaps following Pliny 's description of the dark varnish used by Apelles . Bibliography ( edit ) Editions ( edit ) 1435 / 6 , Della Pittura ( in the Tuscan dialect of Italian ) 1439 -- 41 , De pictura ( in Latin ) Translations and Critical editions ( edit ) Spencer , J ( 1956 , 1966 ) On Painting . New Haven . Grayson , Cecil ( 1972 ) On Painting and On Sculpture : The Latin texts of De Pictura and De Statua . Phaidon . Rocco , Sinisgalli ( 2011 ) On Painting . A New Translation and Critical Edition . Cambridge University Press . References ( edit ) Jump up ^ O'Connor , J.J. ; Robertson , E.F. ( January 2003 ) . `` Mathematics and art - perspective '' . University of St Andrews . Retrieved 21 October 2016 . ^ Jump up to : Wright , D.R. Edward ( 1984 ) . `` Alberti 's De Pictura : Its Literary Structure and Purpose '' . Journal of the Warburg and Courtauld Institutes . 47 : 52 -- 71 . JSTOR 751438 . Jump up ^ Edgerton , Samuel Y. ( October 2006 ) . `` Brunelleschi 's mirror , Alberti 's window , and Galileo 's ' perspective tube ' '' . História , Ciências , Saúde - Manguinhos. 13 ( suppl. 0 ) . doi : 10.1590 / S0104 - 59702006000500010 . Jump up ^ Blunt , Anthony , Artistic Theory in Italy , 1450 - 1660 , ( 1985 ) ( 1940 ) OUP , ISBN 0 - 19 - 881050 - 4 , pages 11 -- 12 Jump up ^ Barlow , Paul , `` The Death of History Painting in Nineteenth - Century Art ? '' PDF , Visual Culture in Britain , Volume 6 , Number 1 , Summer 2005 , pages 1 -- 13 Jump up ^ Alpers , Svetlana Leontief ( July 1960 ) . `` Ekphrasis and Aesthetic Attitudes in Vasari 's Lives '' . Journal of the Warburg and Courtauld Institutes . 23 ( 3 / 4 ) : 190 -- 192 . JSTOR 750591 . Jump up ^ Lefaivre , Liane ( 2005 ) . Leon Battista Alberti 's Hypnerotomachia Poliphili : Re-Cognizing the Architectural Body in the Early Italian Renaissance . MIT Press . pp. 135 -- . ISBN 978 - 0 - 262 - 62195 - 3 . ^ Jump up to : Ackerman , James S. ( 1994 ) . Distance Points : Essays in Theory and Renaissance Art and Architecture . MIT Press . pp. 86 -- 87 . ISBN 978 - 0 - 262 - 51077 - 6 . No particular precept in the first part of De pictura had as pervasive an impact as the following section on perspective construction ; the specific recommendations on color and shadow were ineffectual ... But the incalculable change that Leonardo wrought was directly dependent on Alberti 's vision of art as a branch ... of optics Jump up ^ Grafton , Anthony ; Most , Glenn W. ; Settis , Salvatore ( 2010 ) . The Classical Tradition . Harvard University Press . p. 214 . ISBN 978 - 0 - 674 - 03572 - 0 . 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Hardy A Mathematician 's Apology George David Birkhoff Aesthetic Measure Douglas Hofstadter Gödel , Escher , Bach Nikos Salingaros The ' Life ' of a Carpet Publications Journal of Mathematics and the Arts Organizations Ars Mathematica The Bridges Organization European Society for Mathematics and the Arts Goudreau Museum of Mathematics in Art and Science Institute For Figuring Museum of Mathematics Related topics Droste effect Mathematical beauty Patterns in nature Sacred geometry Retrieved from `` https://en.wikipedia.org/w/index.php?title=De_pictura&oldid=786764550 '' Categories : 1435 works Art history books Hidden categories : All articles with unsourced statements Articles with unsourced statements from October 2016 Talk Contents About Wikipedia Español Français Italiano Piemontèis Edit links This page was last edited on 21 June 2017 , at 13 : 42 . Text is available under the Creative Commons Attribution - ShareAlike License ; additional terms may apply . 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Question: published a treatise on perspective della pitture (or on painting) in 1435