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Dr. Math FAQ:
Browse High School Triangles and Other Polygons
Stars indicate particularly interesting answers or
good places to begin browsing.
Selected answers to common questions:
Area of an irregular shape.
Pythagorean theorem proofs.
- Regular and Non-regular Polygon Areas [03/10/1999]
Given a regular and a non-regular polygon with the same perimeter, prove
that the area of the regular polygon will always be greater.
- A Regular Nonagon [11/2/1995]
I want to know if there is such a thing as a regular nonagon, and if not,
why can't you get one?
- Regular Octagon Problem [09/03/2005]
Find the length of the sides of a regular octagon if the distance
between opposite sides is 20 feet.
- Regular Pentagon Construction Proof [11/23/2001]
What is the proof of the construction of a regular pentagon?
- Regular vs. Equilateral Polygons [07/24/2003]
What is the difference between a regular polygon and an equilateral
- Remembering Area Formulas [12/23/2001]
Is there was a good way to help me memorize the formulas for areas of
- Rhombus and Square Comparison [01/14/2004]
Comparison of the definitions of rhombus and square as a way to answer
the questions, 'Is a square a rhombus?' and 'Is a rhombus a square?'.
- Rhombus vs. Rhomboid [08/27/2002]
What is the difference between a rhombus and a rhomboid?
- Right Angle [04/09/1997]
In a rectangle, draw a line from one vertex to a side to an adjacent
vertex. Determine what makes the angle formed in this process 90 degrees.
- Right Angles in Polygons [07/23/1997]
Is there a relation between the number of sides in a polygon and the
maximum number of right angles?
- Right Triangle Inscribed in a Parabola [09/20/1999]
Show that the point of intersection Q of the axis of the parabola y=x^ 2
and the hypotenuse of right triangle RST (inscribed in the parabola so
that R coincides with the vertex of the parabola) is independent of the
choice of right triangle.
- A Right Triangle of Points [01/14/1999]
Determine the values of x that would make the points (x,0), (-2,1), and
(3,4) the vertices of a right triangle.
- Right Triangle Proof [11/19/2004]
In right triangle ABC, let CD be the altitude to the hypotenuse. If
r1,r2,r3 are radii of the incircles of triangles ABC, ADC, and BDC,
respectively, prove CD = r1 + r2 + r3.
- Right Triangles [9/22/1995]
How do you figure the angle of a right triangle when you only have the
height and width?
- Roof Rafters [02/03/1997]
What is the formula that gives the length of a roof rafter if only the
pitch of the roof is known?
- Round Robin Tournament Schedule [03/31/2000]
Is there a systematic way to come up with a schedule for a round robin
tournament for up to 32 teams, where each team plays every other team
- SAT Rectangle Area Question [04/23/2004]
In rectangle ABCD, Point E is the midpoint of side BC. If the area of
quadrilateral ABED is 2/3, what is the area of rectangle ABCD?
- Scaling a Right Triangle [12/01/2003]
Start with a right triangle of known dimensions. Move the hypotenuse
by some distance towards the right angle. What are the dimensions of
the new triangle?
- Secant-Tangent Theorem [03/21/2002]
I'm trying to prove the secant-tangent theorem.
- Shortest Triangle Side [08/10/2002]
A triangle has a base of length 13, and the other two sides are equal
in length. If the lengths of the sides are integers, what is the
shortest possible length of a side?
- Side Length of a 17-gon [03/03/2002]
Given a 17-gon inscribed in a circle of radius 1, what is the length of a
side to 6 decimals?
- Side Length of Octagon Inscribed in Square [9/11/1996]
I have a piece of square plywood (48" on a side) and I would like to cut
an octagon out of the middle. What is the side length of the octagon?
- Side Lengths of Isosceles Triangle [7/8/1996]
Given two isosceles triangles on top of one another... find the unknown
- Sides of a 30-60-90 Triangle [03/13/1998]
In a 30-60-90 triangle where the short side is X, why does the hypotenuse
equal 2X and the long side equal X * sqrt(3)?
- Similar Triangles [1/22/1996]
For triangle ABC whose vertices are A(6,3),B(1,5),C(-1,4), what are the
vertices of a similar triangle whose perimeter is 5 times that of
- Similar Triangles and Area [11/17/1998]
P is a point on the segment joining midpoints D, E of the sides AB, AC of
a triangle ABC. Prove that BPC has twice the area of ADE.
- Simson Line [04/19/1999]
What is the Simson line?
- Simson Lines [06/07/2001]
Show that, given two triangles inscribed in the same circle, for any
point P on the circle the two Simson's lines form a fixed angle.
- Simson/Wallace Line Proof [11/19/2002]
From a point P on the circumcircle of the triangle ABC perpendiculars
are dropped to the sides AB, BC, CA. Prove that the line joining the
feet of the perpendiculars bisects the line joining the orthocentre of
triangle ABC and point P.
- Sine, Co-sine, and Tangent: SOHCAHTOA [03/28/1999]
I am having trouble figuring out what to use when solving a triangle
- Sine of 36 Degrees [11/18/2001]
Ptolemy calculated the sine of 36 degrees geometrically using the
construction of a regular pentagon. How did he do it?
- Sixteen-sided Window [08/05/1997]
I would like to make a sixteen-sided window for the second floor hallway
of my twin girls' playhouse.
- Spherical Geometry and Triangles [02/09/2004]
Is it possible to have a triangle with two 90 degree angles, where the
other two legs from the connected 90 degree angles meet to finish the
triangle? Where would you find such a triangle? I thought it might
work if the triangle is on a sphere, but then the lines aren't straight.
- The Spider and the Fly [12/23/1999]
A spider and a fly are on opposite walls of a rectangular room... Does
the spider get the fly?
- Spreadsheet to Prove that A = pi*r^2 [3/18/1995]
I need to prepare a spreadsheet using repetitive calculations to prove
that A = pi*r^2. Help!
- Square Inscribed in a Circle [09/28/1997]
What percent of the circle is contained within the square?
- Square Inside a Square [01/30/2001]
Imagine a square with eight compass points marked at each corner and
midpoints of the sides. Create a smaller square inside... How do the
areas of the two squares compare, and why?
- Square Peg, Round Peg [08/22/1997]
Which fits better, a square peg in a round hole, or a round peg in a
- Squares and Circles: How Many Intersections? [02/27/2002]
What is the largest possible number of times a square can intersect a
circle when the square is placed on top?
- Square with Area Equal to a Triangle [10/23/1999]
Given an arbitrary triangle, how can you construct a square with the same
area as the triangle?