Basic Math and Pre-Algebra For Dummies (81 page)

The formula for the area
(A)
of a circle also uses π:

Here's how to use this formula to find the approximate area of a circle with a radius of 5 mm:

In three dimensions, the concepts of area has to be tweaked a little. Recall that, in 2-D, the area of a shape is the measurement of what's inside the shape. In 3-D, what's inside a solid is called its
volume
.

 The
volume (V)
of a solid is a measurement of the space it occupies, as measured in cubic units, such as cubic inches (in.
³
), cubic feet (ft.
³
), cubic meters (m
³
), and so forth. (For info on measurement, flip to Chapter
15
.)
Finding the volume of solids, however, is something mathematicians love for you to know. In the next sections, I give you the formulas for finding the volumes of a variety of solids.

The main measurement of a cube is the length of its side
(s)
. Using this measurement, you can find out the volume of a cube, using the following formula:

So if the side of a cube is 5 meters, here's how you figure out its volume:

You can read 125 m
³
as “125 cubic meters” or, less commonly, as “125 meters cubed.”

The three measurements of a box (or rectangular solid) are its length
(l)
, width
(w)
, and height
(h)
. The box pictured in Figure 
16-18
has the following measurements:
l
= 4 m,
w
= 3 m, and
h
= 2 m.

You can find the volume of a box, using the following formula:

So here's how to find the volume of the box pictured in this section:

Finding the volume of a prism (see prisms in Figure 
16-7
) is easy if you have two measurements. One measurement is the
height (h)
of the prism. The second is the
area of the base (A
_b
)
. The
base
is the polygon that extends vertically from the plane. (In “2-D: Measuring on the flat,” earlier, I show you how to find the area of a variety of shapes.)

Here's the formula for finding the volume of a prism:

For example, suppose a prism has a base with an area of 5 square centimeters and a height of 3 centimeters. Here's how you find its volume:

Notice that the units of measurements (cm
²
and cm) are also multiplied, giving you a result of cm
³
.

You find the volume of cylinders the same way you find the area of prisms — by multiplying the area of the base
(A
_b
)
by the cylinder's height
(h)
:

Suppose you want to find the volume of a cylindrical can whose height is 4 inches and whose base is a circle with a radius of 2 inches. First, find the area of the base by using the formula for the area of a circle: