**How to Measure in Three Dimensions**

Measuring in three dimensions is similar to measuring in two dimensions; however, in 3-D, the boundary of a solid is called its *surface area* (not its perimeter)* *and what’s inside a solid is called its *volume* (not its area).

The surface area of a solid is a measurement of the size of its surface, as measured in square units such as square inches (in.^{2}), square feet (ft.^{2}), square meters (m^{2}), and so forth. 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.

*Measuring spheres*

*Measuring spheres*

The center of a sphere is a point that’s the same distance from any point on the sphere itself. This distance is called the *radius* (*r*) of the sphere. If you know the radius of a sphere, you can find out its volume using the following formula:

Because this formula includes p, using 3.14 as an approximate value for p gives you an approximation of the volume. For example, here’s how to figure out the approximate volume of a ball whose radius is 4 inches:

(**Note:** In the preceding problem, you use equal signs when a value is equal to whatever comes right before it and approximately-equal-to signs () when you round.)

*Measuring cubes*

*Measuring cubes*

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:

*V* = *s*^{3}

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

*V* = (5 m)^{3} = 5 m 5 m 5 m = 125 m^{3}

You can read *125 m** ^{3}* as

*125 cubic meters*or, less commonly, as

*125 meters cubed*.

*Measuring boxes (rectangular solids)*

*Measuring boxes (rectangular solids)*

The three measurements of a box (or rectangular solid) are its length (*l*), width (*w*), and height (*h*). The box pictured in the figure below 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:

*V** = **l** ** **w** ** **h*

So here’s how to find the volume of the box pictured above:

*V* = 4 m 3 m 2 m = 24 m^{3}

*Measuring prisms*

*Measuring prisms*

Finding the volume of a prism 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.

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

*V = A*_{b}* ** h*

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:

*V* = 5 cm^{2} * *3 cm = 15 cm^{3}

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

*Measuring cylinders*

*Measuring cylinders*

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*):

*V** = **A*_{b}* ** **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:

*A** _{b}* = p

*r*

^{2}

3.14 (2 in.)^{2}

= 3.14 4 in.^{2}

= 12.56 in.^{2}

This area is approximate because you use 3.14 as an approximate value for p.

Now use this area to find the volume of the cylinder:

*V* 12.56 in.^{2} 4 in. = 50.24 in.^{3}

Notice how multiplying square inches (in.^{2}) by inches gives a result in cubic inches (in.^{3}).

*Measuring pyramids and cones*

*Measuring pyramids and cones*

The two key measurements for pyramids and cones are the same as those for prisms and cylinders: the height (*h*) and the area of the base (*A** _{b}*). Here’s the formula for the volume of a pyramid or a cone:

For example, suppose you want to find the volume of an ice cream cone whose height is 4 inches and whose base area is 3 square inches. Here’s how you do it:

Similarly, suppose you want to find the volume of a pyramid in Egypt whose height is 60 meters with a square base whose sides are each 50 meters. First, find the area of the base:

*A*_{b}* = **s** ^{2}* = (50 m)

^{2}= 2,500 m

^{2}

Now use this area to find the volume of the pyramid: