See also the
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.
- 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?
- Square with same Perimeter and Area as a Triangle [10/16/1996]
I've been hunting for a square/triangle combination with the same perimeter and area. Is this possible?
- SSA and Non-congruent Triangles [08/13/1998]
Why can't you conclude that two triangles are congruent when side- side-
angle are congruent?
- SSA Proof [03/14/2001]
Proving congruence using the Side-Side-Angle Theorem.
- SSA Theorem: Valid or Invalid? [12/19/2001]
Why can't the SSA Theorem be used to prove congruence?
- SSS, ASA, SAS Proofs [01/21/2002]
I understand the ideas, but I'm not sure when and where to use them.
- Stars in a Flag [04/15/1999]
Find the area of the stars in the American Flag.
- Steiner-Lehmus Theorem [01/28/1999]
I have a proof about an isosceles triangle that I just can't figure
- Steiner-Lehmus Theorem [03/24/2000]
If the lengths of two angle bisectors of a triangle are equal, prove that
it is an isosceles triangle; if the opposite angles of a quadrilateral
add up to 180 degrees, prove that it is a cyclic quadrilateral.
- Stewart's Theorem [5/18/1996]
I have to give a lesson/report on the history and uses of Stewart's
- Straightedge and Compass Constructions [12/14/1998]
Can you help me with these constructions, using only a straightedge and a
compass? A 30, 60, 90 triangle, the three medians of a scalene
- Subsets of Shapes [01/27/2004]
What is the relationship between square and a rectangle?
- Summing Odd Numbers Geometrically [10/30/1999]
Can you prove that 1 + 3 + 5 + ... + (2n-1) = n*n by using a simple
- Sum of Angles of Polygon... [9/24/1996]
Assuming the equality of alternate interior angles formed by a
transversal cutting a pair of parallel lines, prove...
- Sum of Degrees in a Triangle [03/03/1999]
Four proofs that the degrees in a triangle sum to 180.
- Sum of Interior and Exterior Angles [02/14/2003]
Is there a theorem for concave polygons about the sum of the interior
and the sum of the exterior angles?
- Sum of Star Angles [12/19/2001]
Find the sum of the measure of the angles formed at the tips of each irregular star.
- Sum of the Angles in an N-Pointed Star [11/29/1999]
Can you tell me how to find an equation for the sum of the angles in the
tips of an n-pointed star?
- Surveyor's Formula [06/05/1999]
Can you give me a method to calculate the area of an irregular polygon
given all the coordinates of the points?
- Symmedian Point [11/18/2002]
Prove that in the plane of any triangle ABC, with G the centroid, La,
Lb, and Lc the bisectors of angles A, B, and C, Ga, Gb, and Gc the
reflections of line AG about La, BG about Lb, and CG about Lc, the
three lines Ga, Gb, Gc meet in the symmedian point.
- Tangent Line and Circles [04/05/1999]
Two circles of different radius are tangent to each other. A line is
drawn tangent to both circles. How long is the segment between the two
points of tangency of the line and the circles?
- Teaching about Bearings [06/08/2000]
What are bearings? Do you have any ideas on how I can present bearings to
my math class in an interesting fashion?
- Tessellation [02/26/1998]
Are there any non-regular convex polygons with more than four sides that
- Theorems for Quadrilaterals [11/12/1999]
Methods of proving congruence of quadrilaterals similar to the ASA, SAS,
SSS congruence postulates for triangles.
- Three Facts Necessary to Find a Triangle [09/13/2002]
Is there a formula for solving for a right triangle given only the
length of one leg and no angles except for the known 90-degree angle?
- Three Pieces of a Stick Forming a Triangle [01/22/2007]
If you break a straight stick into three pieces, what is the
probability that you can join the pieces end-to-end to form a triangle?
- Ticking Off Congruence [02/06/2013]
A teacher's textbook, and his colleagues, all assume that if two geometric objects have
different tick marks, then the two angles or segments indicated must be incongruent.
Doctor Peterson unpacks the ambiguity, then warns against the larger error of reading
too much in sketches.
- Tiling a Floor [06/30/1999]
How many square yards is a 12ft. by 15ft. room? How many 8" x 8" tiles
would you need for a 30 sq. ft. room?
- Total Area of Multiple Objects [07/18/2002]
I have 6 windows and want to know the total square feet. Do I compute
the areas separately, or add the dimensions and use those to compute
the area all at once?
- Trapezoid Median [8/14/1995]
PQRS is a trapezium with PQ parallel to SR. If A and B are mid-points of
SP and RQ respectively prove that...
- Trapezoid: Visual Proof of Area Formula [04/11/1998]
How can I prove visually that the area of a trapezoid is half the sum of
the parallel sides times the height?
- Triangle Altitude and Area [03/07/1999]
Using Heron's formula to find the altitude of a triangle whose dimensions
- Triangle Altitudes [03/05/1999]
Prove that the three altitudes of a triangle intersect in a common point.
- Triangle and Interior Point [03/23/1999]
Let P be a point inside triangle ABC. AP, BP and CP meet three sides BC,
CA and AB at R, S and T, respectively...
- Triangle Area [08/13/1997]
How do you find the area of a triangle?
- Triangle Area Proofs [01/23/2002]
An analytic proof.
- Triangle Centers at Lattice Points [09/03/2002]
Is there a triangle that can be plotted on a rectangular grid so that
all of its vertices and all four centers are lattice points? If so,
what are the coordinates of the vertices?