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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Pi and Polygons [03/14/1999]

Derive a formula to find the angle of an n-sided polygon with x sides.

Pick's and Euler's Theorems [05/06/1999]

What is Pick's theorem and how can it be linked with Euler's theorem?

Pick's Formula [05/22/1997]

What is Pick's Formula?

Pick's Theorem [9/4/1996]

Could you explain Pick's Theorem?

Picture Frame, Triangle Measurements [5/20/1996]

My teacher gave us ten questions to answer and I could do all except two: 1) A framed rectangular picture is 35cm long and 25cm wide... 2) The base of a triangle is 9cm more than the perpendicular height...

Picture of Icosahedron [7/31/1996]

Do you have a picture of an icosahedron?

Pipe Needed for Natural Gas Line [09/03/1997]

Two houses are 130 metres apart at distances of 55 and 105 metres...

Pixels in a Triangle [08/02/1999]

How can I find the number of pixels inside a triangle?

Placing Coins That Touch [8/7/1996]

How many 20-cent coins can you put around a 20-cent coin so that all of them touch?

A Point Inside a Square [09/18/2003]

Point P lies inside square ABCD such that P's distance from A is 1, P's distance from B is 4 and P's distance from C is 5. What is the area of the square?

Point Inside or Outside Triangle? [08/02/2001]

I have coordinates of three vertices of the triangle and coordinates of the point.

A Point in the Triangle [02/12/1999]

Finding the point P in a plane of triangle ABC, where PA + PB +PC is minimum.

Point within a Triangle [05/29/2003]

I have the coordinates of the three corners of a equilateral triangle ABC. How can I decide whether an arbitrary point (X,Y) lies in the plane of the triangle?

Pole in a Box [02/09/1999]

Can a pole 6.5m long fit into a truck with dimensions of 3m, 3.5m, and 4m?

Polygon Algorithms [05/10/2001]

Given a polygon as a set of points (X, Y) and a database table with X and Y columns, select all records/points from the table that are inside the polygon or belong to its border.

Polygon Diagrams [05/11/2003]

I would like to see pictures of polygons with 11, 12, 13, 14, and 15 sides.

Polygon Names II [12/11/2003]

Why is the triangle named "triANGL"', unlike all the other polygons, which have names like "quadriLATERAL" or "pentaGON"?

Polygons and Triangles [03/09/1999]

Prove that after splitting a regular n-polygon into n triangles, the isosceles triangles have greater area than the scalene triangles.

Polygons, Infinite Sides, and Circles [04/03/1997]

Can a regular polygon with an infinite number of sides be a circle?

Polyominoes [09/08/1997]

I am using polyominos, but I do not know how to tell my dad what they are. How can I tell him so he will know?

A Practical Use for the Orthocenter [03/07/2001]

Does the orthocenter of a triangle have any practical uses?

Probability in the Infinite Plane [03/29/2003]

Three randomly drawn lines intersect so as to form a triangle on an infinite plane. What is the probability that a randomly selected point will fall inside that triangle?

Probability of Making Triangle with Three Stick Pieces [02/24/2007]

Take a stick of unit length and break it into two pieces, choosing the break point at random. Now break the longer of the two pieces at a random point. What is the probability that the three pieces can be used to form a triangle?

Proof by Contradiction [09/25/1997]

Prove that no isoceles right triangle exists which has all three sides integers.

Proof: Median of a Trapezoid Theorem [03/24/2001]

Prove that the median of a trapezoid is: 1) parallel to its bases; 2) length = 1/2 the sum of the bases.

Proof of Congruency [10/13/1996]

Line PR bisects angles QPS and QRS; prove that segments RQ and RS are congruent.

Proof of Hero's formula [09/08/1997]

Could you tell me where to find a proof of Hero's formula or help on how to derive it?

Proof of Morley's Theorem [08/09/2000]

How can I prove Morley's theorem (if every angle in a triangle is trisected, each pair of trisectors meets in a point, and all three points form the vertices of an equilateral triangle)?

Proof of Pappus' Theorem [04/02/2001]

How can I prove Pappus' theorem of colinearity with the help of Menelaus' theory?

Proof of Pythagorean Theorem: Pythagoras' Reasoning [02/23/1998]

I would like to know how Pythagoras reasoned his theorem.

Proof of the Feuerbach Theorem [03/14/2000]

Please submit the proof of the Feuerbach theorem (the nine-point circle is tangent to the incircle and the circumcircle of a triangle.)

Proof of the Parallelogram Law [08/07/1999]

How do you prove the parallelogram law geometrically, without using vectors?

Proofs with Isosceles Triangles [10/28/1998]

What are altitudes, angle bisectors, and medians? How do you prove that in an isoseles triangle, the altitude is a median and an angle bisector?

Proof Using Coordinate Geometry [06/08/2001]

In equilateral triangle ABC, a segment is drawn from point A to the side BC at a point (D)... prove that angle BFC (or EFC) is a right angle.

Properties of Equilateral Triangles [07/20/1998]

If ABC is equilateral and AD is one of its heights, what are the measures of the angles? Is ADB equal to ADC? If AB = 2 find BD and AD.

Prove Triangle of Sides with Length... [8/1/1996]

Let a, b and c be the lengths of the sides of a triangle. Prove that the square root of (a+b-c) plus the square root of (b+c-a) plus the square root of (c+a-b) is equal to or less than the sum of the square roots of a, b and c. Determine when equality occurs.

Prove Triangles are Similar [12/16/1995]

How do you prove two triangles are similar?

Proving Lines Congruent [03/29/2002]

Prove line AL is congruent to line CM.

Proving Quadrilateral is a Parallelogram [11/30/2001]

We are having a problem with the idea of a quadrilateral having one pair of opposite sides congruent and one pair of opposite angles congruent.

Proving the Diagonals of a Rectangle Congruent [12/6/1995]

How would you prove that the diagonals of a rectangle are congruent?

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