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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Area of an Octagon [10/26/2001]

I am trying to figure out the square footage of an octagon-shaped house. Each wall measures 15 ft. in length.

Carrying a Ladder around a Corner [02/28/2003]

A ladder of length L is carried horizontally around a corner from a hall 3 feet wide into a hall 4 feet wide. What is the length of the ladder?

Congruence and Triangles [12/13/1997]

Can you please explain how to determine, using SSS, SAS, and ASA, how a shape is congruent or not?

General Area Formula [02/14/2002]

Is there an all-inclusive formula for the area of a square, rectangle, parallelogram, trapezoid, and triangle?

Polygon Angles [02/14/1997]

What is the sum of the measure of the angles in polygons with sides 3-50?

Polygon Diagonal Formula [8/20/1996]

Does the polygon diagonal formula apply to other parts of geometry?

Possible Areas of a Triangle [12/27/2001]

Exploring the areas of a triangle with side lengths 6 and 7.

Pythagorean Proof Based on the Principles of Scaling [04/04/2002]

I've decided to do a project with some connections to the Pythagorean theorem, but the project requires innovative ideas.

The Pythagorean Theorem [07/07/1997]

Could you please explain the Pythagorean Theorem?

Pythagorean Theorem and non-Right Triangles [03/09/2002]

Why doesn't the Pythagorean theorem work for triangles other than right triangles?

Pythagorean Theorem: Why Use the Converse? [07/15/2003]

Why use the converse of the Pythagorean theorem?

Theorem or Postulate? [11/03/2002]

Shouldn't the three triangle postulates - SSS, ASA, and SAS - be theorems?

Triangle Congruence [05/13/2003]

I don't understand how to tell if two triangles are congruent.

Two Crossing Ladders [07/01/2003]

Two walls are 10 ft. apart. Two ladders, one 15 ft. long and one 20 ft. long, are placed at the bottoms of the walls leaning against the opposite walls. How far from the ground is the point of intersection?

16-sided Regular Polygon [07/31/2001]

How can I construct a 16-sided polygon?

30-60-90 and 45-45-90 Triangles [03/15/1999]

If I have a triangle that is 30-60-90 or 45-45-90, how do I find all the sides when given only one side? Where does trigonometry come in?

8 Sticks, No Triangle [05/12/2003]

No triangle can be formed using any three of 8 sticks, each of integer length. What is the shortest possible length of the longest of the eight sticks?

AAA, ASS, SSA Theorems [11/16/2001]

Why can't AAA, ASS, and SSA be used to determine triangle congruence?

Acute Angles in a Triangle [12/02/1998]

What is the greatest number of angles smaller than a right angle that a triangle can have?

Altitudes and Bisectors of a Triangle [05/25/1999]

Prove that the altitudes of a triangle are bisectors in the triangle formed by connecting the meeting points of the altitudes with the sides of the original triangle.

The Ambiguous Case [04/01/2003]

How many triangles can be constructed if, for example, a=4, A=30, and c=12? Or a=9, b=12, and A=35?

Angle Between Two Sides of a Pyramid [10/29/1999]

How can I compute the angle formed by two sides of a frustrum of a pyramid?

The Angle Bisector and Equal Side Ratios [05/17/1998]

Given triangle ABC and angle bisector BD, show that AB/AD = BC/CD.

Angle-bisector Proof [10/16/1997]

Prove that in a triangle ABC, a pair of angle-bisectors cannot be perpendicular.

Angle Bisector Theorem [10/11/2002]

The bisector of an interior angle of a triangle divides the opposite side internally into two segments that are proportional to the adjacent sides.

Angle Measurements of Triangles inside Semicircle [11/26/1998]

If the area of a triangle inside a semicircle is equal to the area outside the triangle within the semicircle, then find the values of the acute angles in the triangle.

Angle of Elevation [01/22/1997]

A tree 66 meters high casts a 44-meter shadow. Find the angle of elevation of the sun.

Angle, Side Length of a Triangle [9/4/1996]

What is the relation between the angles and side lengths of a triangle?

Angle-Side-Side Does Not Work [11/12/2001]

Can you give me a construction to show that Angle-Side-Side does not prove two triangles congruent?

Angles of a Cyclic Quadrilateral [07/14/1998]

ABCD is a cyclic quadrilateral with AB parallel to DC. Angle DAC = 40 degrees...

Angles of a Triangle [02/04/2003]

Why do the angles of a triangle always add up to 180 degrees?

Angles of Stars [08/18/1997]

What are the interior and external angles of stars built on regular pentagons and octagons.

Another Isosceles Triangle [10/27/1999]

A triangle has sides of length 29, 29, and 40 cm. How can I find another isosceles triangle with the same perimeter and area that also has sides of integral length?

Ant and Rectangle [01/22/2001]

Does the ant walk along the diagonals of the rectangle?

Apothem of a Hexagon [6/11/1996]

What is the formula for the apothem of a regular hexagon?

Apothem of a Triangle [03/21/2001]

Find the apothem and radius of a triangle with a side of length 12.

Applying Euler's Methods [07/27/1999]

Questions about prime divisors, triangle constructions, decomposing quartic polynomials, and rational roots.

Approximating Pi using Geometry [08/12/1998]

I need to know a simple method to find the approximate value of pi using elementary geometry.

Arcs Inside a Square [07/25/1999]

What is the area of the figure created by the intersection of two arcs drawn in a square of sidelength 5 units?

Area and Perimeter in Polygons [06/24/1999]

How can I prove the formula A = (a^2n)/(4tan(180/n)) for computing the area of a regular n-gon with sidelength a? How does this compare to the area of a circle?

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