Ask Dr. Math

High School Archive

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.
    Classifying quadrilaterals.
    Heron's formula.
    Polygon diagonals.
    Pythagorean theorem proofs.
    Triangle congruence.

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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 triangles?

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 diagonals?

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/ AD)?

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, find PB.

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 is correct?

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 definition.)

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 triangle?

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.

Determining Triangle Similarity [05/26/1998]

Given two triangles, how can you determine if they are similar?

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