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.
- Constructing a Segment [09/26/1999]
Given a 1" segment and a 2.5" segment, how can you find a segment of
length sqrt(2.5)" using only a compass and a straightedge?
- Constructing a Square [12/25/1998]
Given any four points, construct a square such that each side or
extension passes through one point.
- Constructing a Triangle [08/20/1999]
How can you construct a triangle with 3 different-size segments?
- Constructing a Triangle Given the Medians [01/01/2001]
How can I construct a triangle ABC given AM, BN, and CP, the respective
medians from the vertices A, B, and C?
- Constructing Polygons [06/03/1998]
How do you construct a regular pentagon and a regular decagon? Can you
construct a regular n-gon?
- Constructing the Orthocenter [01/27/1999]
How do you construct the orthocenter of a triangle?
- Construct Polygon Given One Side [12/03/2001]
How can you construct a polygon, given one side?
- Converse of the Pythagorean Theorem [02/14/2003]
What is the converse of the Pythagorean theorem?
- Cosine Addition Formula [12/13/1997]
How can you prove the addition formula for cosine by using right
- Counting Collinear Sides [04/12/2013]
A student wonders how to count the sides of a polygon that has degenerate vertices.
Citing Wikipedia and MathWorld, Doctor Peterson teases out the various conventions.
- Counting Diagonals [03/14/1998]
How many diagonals can be drawn for a polygon with n sides?
- Counting Intersections of Diagonals in Polygons [03/08/2000]
Can you help me find an equation for the maximum number of intersections
of the diagonals in a polygon?
- Counting Rectangles Cut By a Diagonal [06/15/1999]
How can we find an equation for the number of unit squares that are cut
by a line going from corner to corner on a rectangle?
- Counting Sides by Counting Diagonals [06/04/2002]
How can I find the number of sides in a polygon, given the number of
- Covering Paper using Index Cards [10/24/2001]
What is the maximum area of an 8"x13" sheet of paper that you can cover
by using seven 3"x5" standard index cards?
- Cross-Cornering a Shape to Make it Square [12/19/2002]
We lay out a building that is 30' x 40' and cross-corner it to see if
it is 'square,' but there is a 6' difference. What is the equation to
find how far to move one side to make the shape 'square'?
- Curious Property of a Regular Heptagon [04/06/2001]
How can I prove that in a regular heptagon ABCDEFG, (1/AB)=(1/AC)+(1/
- Cutting a Circle out of a Square [2/14/1996]
What is the area (to the nearest square centimeter) of the largest circle
that can be cut from a square piece of sheet metal 73cm. on each side?
Explain how you determined this.
- Cutting a Square into Five Equal Pieces [07/12/1999]
How can you divide a square cake into five equal parts, cutting through
the center point?
- Cutting a Triangle into Two Congruent Triangles [10/06/1998]
How do you cut a triangle into two congruent equilateral triangles with
the minimum number of cuts?
- Cutting Carpet [9/9/1996]
Two pieces of carpet are to be used to cover a floor. You are allowed to
make just one cut in one of the two pieces...
- Cyclic Quadrilateral [05/22/2000]
For an isoscles trapezium ABCD with AB paralled to DC and AB less than
CD, how can we prove that ABCD is a cyclic quadrilateral?
- Cyclic Quadrilaterals [8/30/1996]
A cyclic quadrilateral touches a circle at each vertex. What angles do
these points make with the centre of the circle?
- Cyclic Trapezoid [01/17/2002]
PQ is a diameter; AB is a chord parallel to PQ. If PQ=50cm and AB= 14cm,
- Defining Exterior Angles of Polygons [06/24/2004]
My math teacher says the sum of the exterior angles of a triangle is
900 degrees (360*3 - 180). I think that the sum is 360 degrees. Who
- Definition of Opposite Sides [01/18/2001]
What is the formal definition of 'opposite sides' of a polygon? Does a
regular pentagon have opposite sides? Does a concave polygon have
opposite sides? How can we define it consistent with our intuition?
- Degenerate/Nondegenerate Figure [10/27/2001]
We need to know what a nondegenerate circle is. (We're trying to decide
whether this is a model of incidence geometry, but don't know the
- Degenerate Triangle [09/03/2003]
Isn't a degenerate triangle really just a line segment?
- De Longchamp's Point [09/21/2000]
What is De Longchamp's point, and how is it used?
- Derivation of Law of Sines and Cosines [11/02/1997]
How do you derive the law of sines and the law of cosines?
- Derivations of Heron's Formula [11/24/1998]
How is Heron's formula (Hero's formula) derived?
- Derivations of Pi and a Polygon of Degree n [3/4/1995]
I am curious about the derivation of pi and the formula for deriving a
polygon of degree n. Can you help?
- Deriving the Distance Formula from the Pythagorean Theorem [03/23/2003]
How does one derive the distance formula from the Pythagorean theorem?
- Deriving the Law of Cosines [04/01/1998]
Will the Pythagorean Theorem work with a non-right triangle?
- Deriving Trilinear Coordinates [05/18/1999]
How do you derive the trilinear coordinates of the orthocenter of a
- Desargues' Theorem and SSASS [12/15/1998]
What is the main theory behind Desargues' Theorem? Also, is SSASS a valid
method for proving two quadrilaterals are congruent?
- Determinants and the Area of a Triangle [12/14/1998]
Given a triangle with vertices (A,B), (C,D), and (E,F), how do you find
the area in determinant form?
- Determine if Point is in Rectangle [5/29/1996]
What formula will allow me to determine whether a specified point lies
within a polygon (rectangle - 4 points)?
- Determining Area of a Lot of Land [08/20/2003]
We are trying to figure out the square footage for a lot that I own.
- Determining If a Given Point Lies inside a Polygon [12/27/2005]
I have a finite number of points that constitute a polygon, and a
point p(x,y). I want to know if point p lies inside the polygon.