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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Do Figures with Equal Sides Have the Same Area? [11/20/2008]

If a rectangle is tipped slightly so the corners are not square but the side lengths don't change, does the area of the figure change or stay the same?

Equable Rectangle [01/28/2001]

What is the formula for an equable rectangle if I have one side of the rectangle?

Equable Shapes [04/26/2001]

Is there a rule for how to get equable shapes, that is, shapes with the same area and perimeter?

Equilateral, Isosceles, Scalene - Word Origins [12/09/2001]

I need to find out about the origins of the scalene, isoceles, and equilateral triangles. How they were named?

Estimating Seating Capacity [05/10/2000]

How can I estimate how many chairs can be placed in the seating area of a convention center meeting room that is 80 feet across at the front, 120 feet across at the rear, and 150 feet from front to back?

Etymology of the Word Hypotenuse [01/26/2003]

Why is the hypotenuse called 'hypotenuse'?

Etymology of the Word Rectangle [02/07/2003]

Where does the word 'rectangle' come from?

Etymology of the Word Rhombus [01/30/2003]

Where does the word rhombus come from?

Euler Line [06/08/1998]

What is the Euler line?

Evaluating a Trigonometric Expression [07/31/1999]

Can you help me evaluate tan (arcsin (3/5))?

Familiar Triangles [01/19/1999]

How do you get the lengths of the sides of a 45-45-90 triangle and a 30- 60-90 triangle?

Faster Than Algebra [05/14/2002]

The sides of a rose garden in the shape of a right triangle are in a ratio of 8:15:17. If the perimeter is 60 feet, what is the length of the shortest side?

A Fibonacci Jigsaw Puzzle [04/29/1999]

Why is the area of our rectangle, formed from a square, 65 when the square's area was 64?

Find a Point 3/8 along the Line [12/04/2002]

Find a point 3/8 from A to B. Given: two endpoints X,Y coordinates. Point A (-2,7) point B (6,-5).

Finding Area and Volume [04/12/2001]

When working with area and volume of triangular shapes, how do I know when to divide the base by 2 and when to divide it by 3?

Finding Areas of Different Polygons [09/02/1997]

Could you please tell me how to work out the area for an equilateral heptagon, octagon, nonagon, decagon, unedecagon, and dodecagon?

Finding Polygon Areas [03/20/1997]

How do I find the area of polygons?

Finding Rhombi in a Rhombus [9/25/1995]

How can I work out a formula for finding how many rhombi there are in a rhombus? (say 2cm*2cm or 3cm*3cm and so on, etc.)

Finding the Dimensions of a Box [10/21/2001]

You want to construct a cardboard box from a cardboard strip that is 8 inches wide. The dimensions of the box are 8"x8"x4". How long does the strip need to be?

Finding the Perimeter and Area [12/01/1998]

I don't understand how to do perimeter and area in shapes such as a parallelogram and trapezoid.

Finding Total Area of Several Rectangles [07/08/2005]

I need to find the total square footage of a lot of rectangular lawns. Do I have to find the area of each lawn and add up all the areas, or can I just add all the lengths and all the widths and make one area calculation based on those two totals?

Find the Angles [9/11/1996]

The first angle of a triangle is eight degrees more than two times the third angle. The second angle is two degrees less than three times the third angle. Find all three angles.

Find the Length of the Third Side [08/26/2003]

Two cars start at the same point, going in opposite directions for six miles; both make a left turn and go 8 more miles. How many miles apart are the cars?

Find the Perimeter [04/09/2002]

I have a group of 13 rectangles arranged into a larger rectangle. I know the area, but I need to find the perimeter.

Finite vs. Rational [9/10/1996]

A right triangle with sides 1 and 2 has a hypotenuse equal to the square root of 5, which is irrational - it carries on to infinity without recurring - but the side length of a triangle must be finite!

Fitting a Picture to a Frame [11/14/1996]

I have a picture frame that needs to have twice as much border on the side of the picture as on the top. If the photo is half the area of the frame, how wide should the borders be?

Five Equal Pieces of a Square Cake [05/22/2001]

Ravina wants to cut a square cake using straight vertical cuts to make five pieces of equal volume. If she makes the first cut from the cake's center to the top left corner, where must she make the other cuts if they all start from the cake's center?

Formula for Area of Any Regular Polygon [03/01/1998]

Area of a regular polygon, given the number of sides and length of a side.

Formula for the Area of a Trapezoid [01/14/2006]

My math teacher gave us the formula for the area of a trapezoid, but I don't understand his explanation of why it works. Can you explain it?

Formulas for the Area of a Figure [3/23/1996]

I am in 8th grade and I have one week left in Geometry and I still can't get how to find the area of a figure!

The Geodesic Dome [8/4/1995]

What polygons make up a geodesic dome and where are they located?

Geometrically Completing the Square [08/07/1997]

What are the steps for geometrically completing the square?

Golden Triangle: An Isosceles Triangle [01/23/2001]

What is the Golden Triangle?

Golden Triangle: What is It? [09/19/1999]

What is a Golden Triangle?

Height of Parallelogram or Trapezoid [04/30/1999]

Could you explain the concept of height with regard to a parallelogram or a trapezoid?

Heron's Formula [04/11/2003]

How can I find the area of an isosceles triangle, an equilateral triangle, a scalene triangle, an obtuse-angle triangle and an acute-angle triangle without the height being given?

Hexagon vs. Hexagram [01/11/1999]

What is the difference between a hexagon and a hexagram?

A Hint for a Geometry Proof [1/6/1995]

Prove: If a segment whose endpoints lie on opposite sides of a parallelogram passes through the midpoint of a diagonal, that segment is bisected by the diagonal.

How Did Eratosthenes Measure the Circumference of the Earth? [6/26/1996]

Didn't Eratosthenes measure the lengths of shadows of sticks at different locations on the same day and time of the year, so he had two right triangles...?

How Many Proofs of the Pythagorean Theorem? [03/27/2003]

Do you know the exact number of proofs of the Pythagorean Theorem in existence?

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