Ask Dr. Math

Middle School Archive

Browse Middle School Triangles and Other Polygons
Stars indicate particularly interesting answers or good places to begin browsing.

Selected answers to common questions:
    Classifying quadrilaterals.
    Polygon diagonals.
    Pythagorean theorem proofs.

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The Area of a Trapezoid [1/27/1996]

My daughter has been assigned to derive the formula for the area of a trapezoid rather than just looking it up.

Finding the Area of Squares and Triangles [06/20/1999]

Could you please explain how to find area?

General Area Formula [02/14/2002]

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

Incenter, Orthocenter, Circumcenter, Centroid [01/05/1997]

I have been having trouble finding the Euler line of a triangle.

Measuring by Shadows [05/22/2001]

How can I measure a tree using its shadow and mine?

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.

The Pythagorean Theorem [07/07/1997]

Could you please explain the Pythagorean Theorem?

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

Why use the converse of the Pythagorean theorem?

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?

A 14-Sided Polygon [7/28/1996]

What is the proper name for a 14-sided figure?

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?

A 78- or 79-Sided Polygon/Polyhedron [3/13/1995]

Is it possible to produce a 78- or 79-sided polygon or polyhedron, and if so, what would it look like?

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?

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

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

Angles of a Parallelogram [8/24/1996]

In a parallelogram ABCD, angle ABC = 2 * angle BCD; what is ADC?

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.

Apothem of a Triangle [03/21/2001]

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

Approximating Pi using Geometry [08/12/1998]

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

Area and Perimeter [05/01/2001]

I do not understand area and perimeter - can you explain them?

Area and Perimeter: Mowing the Lawn [6/1/1996]

How many circuits are necessary to cut half the lawn?

Area of a Half-Mowed Yard [08/21/2003]

There is a square/rectangular yard, size unknown. You have mowed a 10 foot swath around the perimeter. You are now half done. How big is the yard?

Area of an Irregular Polygon [03/29/2001]

How can I find the area of an irregular polygon? What about a polygon made out of rectangles?

Area of a Parallelogram [5/28/1996]

How do you find the area of a parallelogram?

Area of a Parallelogram [01/07/1999]

How can the area of a parallelogram be the same as the area of a rectangle?

Area of a Rectangle Outside a Rectangle [6/11/1996]

Find the area of the concrete border of a rectangular swimming pool.

Area of a Rhombus and Area of a Square [11/16/2007]

A rhombus has four equal sides, so it's a 'pushed over' square. Why is the area formula not side times side like in a square?

Area of a Triangle Explained [06/17/1999]

Why does the formula for the area of a triangle A = (1/2)ab work?

Area of Inscribed Circle [12/01/1998]

Find the area of the circle inscribed in a triangle ABC using Heron's Law.

Areas of Figures Broken into Rectangles [10/17/2001]

Calculate the area of each figure. First divide the figure into rectangles and squares...

Area: Triangle vs. Rectangle [4/30/1996]

Why do you have to use a different formula to get the area of a triangle than a rectangle or square?

Area vs. Perimeter of Rectangles [03/19/2000]

Can you explain how two rectangles with exactly the same perimeter can enclose different areas?

Are Squares Rectangles? Are Rectangles Squares? [03/29/2004]

Can a rectangle be a square? I just was wondering because I know a square can be a rectangle.

Axes of Symmetry [2/3/1996]

Can you help a year 5 student understand axes of symmetry of a triangle?

Base or Width? [06/11/2002]

When computing area, is there a difference between 'width' and 'base'?

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