Idiot\'s Guides Basic Math and Pre-Algebra

232 Part 3: The Shape of the World

WORLDLY WISDOM

Taking the time to sketch out the portions of the figure before you begin calculating

is time well spent. Especially if you’re thinking of the figure as shaded and unshaded

layers, it’s wise to sketch the layers separately.

The smaller white circle has a radius of 3 inches. The gray circle underneath is an inch bigger,

so its radius is 4 inches. The area of the big gray circle is 4^2 times pi, or 16S square inches. The

white circle has an area of 3^2 times pi, or 9S square inches. The part of the gray circle peeking

out from behind the white circle, the gray ring, has an area of 16S – 9S = 7S square inches.

Now that you know the area of that outer ring, you can bring back the gray circle in the middle.

It has a radius of 2 inches, so its area is 4S square inches. Add that to the outer ring, and you have

a shaded area of 7S + 4S = 11S square inches.

You’ve looked at circles in circles, what about squares in squares?

In this figure, the area of square ABCD is 32 square centimeters.

To find the area of the shaded region, work backward to find the

dimensions you need. Square ABCD has an area of 32 square

centimeters, so each of its sides is 32 16 2 4 2 centimeters.

What you need, however, is to find the area of the gray isosceles

triangles.

B

A

D

C

Each side of the square fits the hypotenuses of two triangles, so

each hypotenuse is half of 42 , or 22 centimeters. In an isosceles

right triangle, with two legs the same length, the Pythagorean

theorem says a^2 + b^2 = a^2 + a^2 = 2a^2 = 22

2

. That means that 2a^2
= 8, a^2 = 4 and a = 2, so each leg of the little gray triangles is 2
centimeters long. The area of each little triangle is^12 v 2 v 2 or
2 square centimeters, and there are 8 triangles, making the total
shaded area 16 square centimeters.

WORLDLY WISDOM

Leave any square roots in radical form at least until the end of your calculation. If you

move to a decimal approximation sooner, you’ll have to do your arithmetic with that

decimal. That’s more work, plus it’s approximate to begin with, and you may need to

round more at the end. Wait, keep the radical, and then, if you really need a decimal

approximation, do it when all the calculation is done.