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  #### Resources tagged with Pythagoras' theorem similar to Kissing Triangles:
  
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  Area. Rectangles. Circles. smartphone. Generalising. Mixed triangles. Creating expressions/formulae. Squares. Visualising. Pythagoras' theorem.
  
  ### There are 70 results
  
  Broad Topics > 2D Geometry, Shape and Space > Pythagoras' theorem
  
  ### Tilted Squares
  
  ##### Stage: 3 Challenge Level:
  
  It's easy to work out the areas of most squares that we meet, but what if they were tilted?
  
  ### The Pillar of Chios
  
  ##### Stage: 3 Challenge Level:
  
  Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .
  
  ### Square Pegs
  
  ##### Stage: 3 Challenge Level:
  
  Which is a better fit, a square peg in a round hole or a round peg in a square hole?
  
  ### Pythagorean Triples
  
  ##### Stage: 3 Challenge Level:
  
  How many right-angled triangles are there with sides that are all integers less than 100 units?
  
  ### Floored
  
  ##### Stage: 3 Challenge Level:
  
  A floor is covered by a tessellation of equilateral triangles, each having three equal arcs inside it. What proportion of the area of the tessellation is shaded?
  
  ### Liethagoras' Theorem
  
  ##### Stage: 2 and 3
  
  Liethagoras, Pythagoras' cousin (!), was jealous of Pythagoras and came up with his own theorem. Read this article to find out why other mathematicians laughed at him.
  
  ### Trice
  
  ##### Stage: 3 Challenge Level:
  
  ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED. What is the area of the triangle PQR?
  
  ### Isosceles
  
  ##### Stage: 3 Challenge Level:
  
  Prove that a triangle with sides of length 5, 5 and 6 has the same area as a triangle with sides of length 5, 5 and 8. Find other pairs of non-congruent isosceles triangles which have equal areas.
  
  ### A Chordingly
  
  ##### Stage: 3 Challenge Level:
  
  Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle.
  
  ### Tennis
  
  ##### Stage: 3 Challenge Level:
  
  A tennis ball is served from directly above the baseline (assume the ball travels in a straight line). What is the minimum height that the ball can be hit at to ensure it lands in the service area?
  
  ### The Dangerous Ratio
  
  ##### Stage: 3
  
  This article for pupils and teachers looks at a number that even the great mathematician, Pythagoras, found terrifying.
  
  ### Pythagoras
  
  ##### Stage: 2 and 3
  
  Pythagoras of Samos was a Greek philosopher who lived from about 580 BC to about 500 BC. Find out about the important developments he made in mathematics, astronomy, and the theory of music.
  
  ### Semi-square
  
  ##### Stage: 4 Challenge Level:
  
  What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?
  
  ### Semi-detached
  
  ##### Stage: 4 Challenge Level:
  
  A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius.
  
  ### Compare Areas
  
  ##### Stage: 4 Challenge Level:
  
  Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?
  
  ### Hex
  
  ##### Stage: 3 Challenge Level:
  
  Explain how the thirteen pieces making up the regular hexagon shown in the diagram can be re-assembled to form three smaller regular hexagons congruent to each other.
  
  ### Take a Square
  
  ##### Stage: 4 Challenge Level:
  
  Cut off three right angled isosceles triangles to produce a pentagon. With two lines, cut the pentagon into three parts which can be rearranged into another square.
  
  ### Rhombus in Rectangle
  
  ##### Stage: 4 Challenge Level:
  
  Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.
  
  ### Fitting In
  
  ##### Stage: 4 Challenge Level:
  
  The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest. . . .
  
  ### Squaring the Circle and Circling the Square
  
  ##### Stage: 4 Challenge Level:
  
  If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
  
  ### Napkin
  
  ##### Stage: 4 Challenge Level:
  
  A napkin is folded so that a corner coincides with the midpoint of an opposite edge . Investigate the three triangles formed .
  
  ### The Spider and the Fly
  
  ##### Stage: 4 Challenge Level:
  
  A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?
  
  ### Partly Circles
  
  ##### Stage: 4 Challenge Level:
  
  What is the same and what is different about these circle questions? What connections can you make?
  
  ### Equilateral Areas
  
  ##### Stage: 4 Challenge Level:
  
  ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.
  
  ### Inscribed in a Circle
  
  ##### Stage: 4 Challenge Level:
  
  The area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius?
  
  ### The Old Goats
  
  ##### Stage: 3 Challenge Level:
  
  A rectangular field has two posts with a ring on top of each post. There are two quarrelsome goats and plenty of ropes which you can tie to their collars. How can you secure them so they can't. . . .
  
  ### Star Gazing
  
  ##### Stage: 4 Challenge Level:
  
  Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.
  
  ### Pythagoras Proofs
  
  ##### Stage: 4 Challenge Level:
  
  Can you make sense of these three proofs of Pythagoras' Theorem?
  
  ##### Stage: 4 Challenge Level:
  
  A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?
  
  ### Six Discs
  
  ##### Stage: 4 Challenge Level:
  
  Six circular discs are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box. Can you put the boxes in order according to the areas of their bases?
  
  ### Circle Packing
  
  ##### Stage: 4 Challenge Level:
  
  Equal circles can be arranged so that each circle touches four or six others. What percentage of the plane is covered by circles in each packing pattern? ...
  
  ### Cutting a Cube
  
  ##### Stage: 3 Challenge Level:
  
  A half-cube is cut into two pieces by a plane through the long diagonal and at right angles to it. Can you draw a net of these pieces? Are they identical?
  
  ### Two Circles
  
  ##### Stage: 4 Challenge Level:
  
  Draw two circles, each of radius 1 unit, so that each circle goes through the centre of the other one. What is the area of the overlap?
  
  ### Get Cross
  
  ##### Stage: 4 Challenge Level:
  
  A white cross is placed symmetrically in a red disc with the central square of side length sqrt 2 and the arms of the cross of length 1 unit. What is the area of the disc still showing?
  
  ### Three Cubes
  
  ##### Stage: 4 Challenge Level:
  
  Can you work out the dimensions of the three cubes?
  
  ### Squ-areas
  
  ##### Stage: 4 Challenge Level:
  
  Three squares are drawn on the sides of a triangle ABC. Their areas are respectively 18 000, 20 000 and 26 000 square centimetres. If the outer vertices of the squares are joined, three more. . . .
  
  ### Weighty Problem
  
  ##### Stage: 3 Challenge Level:
  
  The diagram shows a very heavy kitchen cabinet. It cannot be lifted but it can be pivoted around a corner. The task is to move it, without sliding, in a series of turns about the corners so that it. . . .
  
  ### Zig Zag
  
  ##### Stage: 4 Challenge Level:
  
  Four identical right angled triangles are drawn on the sides of a square. Two face out, two face in. Why do the four vertices marked with dots lie on one line?
  
  ### Rectangular Pyramids
  
  ##### Stage: 4 and 5 Challenge Level:
  
  Is the sum of the squares of two opposite sloping edges of a rectangular based pyramid equal to the sum of the squares of the other two sloping edges?
  
  ### Three Four Five
  
  ##### Stage: 4 Challenge Level:
  
  Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.
  
  ### Pareq Calc
  
  ##### Stage: 4 Challenge Level:
  
  Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .
  
  ### Tilting Triangles
  
  ##### Stage: 4 Challenge Level:
  
  A right-angled isosceles triangle is rotated about the centre point of a square. What can you say about the area of the part of the square covered by the triangle as it rotates?
  
  ##### Stage: 4 Challenge Level:
  
  The sides of a triangle are 25, 39 and 40 units of length. Find the diameter of the circumscribed circle.
  
  ### Nicely Similar
  
  ##### Stage: 4 Challenge Level:
  
  If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?
  
  ### Some(?) of the Parts
  
  ##### Stage: 4 Challenge Level:
  
  A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle
  
  ### Corridors
  
  ##### Stage: 4 Challenge Level:
  
  A 10x10x10 cube is made from 27 2x2 cubes with corridors between them. Find the shortest route from one corner to the opposite corner.
  
  ### Round and Round
  
  ##### Stage: 4 Challenge Level:
  
  Prove that the shaded area of the semicircle is equal to the area of the inner circle.
  
  ### All Tied Up
  
  ##### Stage: 4 Challenge Level:
  
  A ribbon runs around a box so that it makes a complete loop with two parallel pieces of ribbon on the top. How long will the ribbon be?
  
  ### Ball Packing
  
  ##### Stage: 4 Challenge Level:
  
  If a ball is rolled into the corner of a room how far is its centre from the corner?