Need help with a geometry problem

[Michelle in MO]

Hello math moms and dads!

 

My oldest and I have been trying to figure out the reasoning for the following geometry problem. It's in the Chalkdust (Larson) text, Section 11.4, Exercise 30.

 

Figuring out the area of a regular octogon:

 

We're supposed to figure out the area two ways:

 

The octogon is divided into two isoceles trapezoids---one on each side, with a rectangle in the middle whose width is also 8 units. The sides of the octogon are all 8 units. (Wish I could draw a picture!)

 

One way: A = 1/2h(b1 + b2); since there are two trapezoids, this would be 2(1/2h[b1 + b2]) PLUS the length times the width of the rectangle in between the two trapezoids. I understand how to do this one.

 

The other way: A = 1/2aP (one half times the apothem times the perimeter). The perimeter is easy: 8X8=64 units. I'm having trouble understanding how the solutions manual came up with the answer for the height of the trapezoid: h = the square root of 8^2 divided by 2; or 4 times the square root of 2. The isoceles trapezoid has a b1 of 8 (the side of the outer perimeter). The height would be not the length of the leg of the trapezoid, obviously, but the height from the lower base, b2, to the upper base, b1. That height seems to form an isoceles right triangle, or perhaps it's even an equilateral right triangle.

 

So, how did they figure out the height of this trapezoid to be h = the square root of 8^2 (which is 64) divided by two, or 4 times the square root of 2? I know how to do the math for that problem, but I guess I can't figure out the reasoning behind this? How do you figure out the height of a trapezoid?

 

Thanks for any and all help!

[Myra]

I tried Google for "height of a trapezoid" - many explanations came up. Here's an esp good one.

 

http://www.sparknotes.com/testprep/books/newsat/chapter20section5.rhtml

 

Hope it helps!

Myra

[Jann in TX]

An interior angle of a regular octagon is 135.

 

Int angle= (n-2)180/n

 

The 'rectangle' is taking up 90 degrees of that so the angle of the trap. is 45 degrees.

 

Draw a 'height' line connecting the b1 to b2 that makes a right triangle. It will be a 45-45-90 triangle. The hypotenuse (8) is sqrt of 2 times a leg (height).

 

Work the math and rationalize the denominator to get the height they listed.

 

I'll e-mail you a picture.