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 (m²), 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 (m³), and so forth.

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

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 = s³

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 m³

You can read 125 m³ as 125 cubic meters or, less commonly, as 125 meters cubed.

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 m³

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 cm² 3 cm = 15 cm³

Notice that the units of measurements (cm² and cm) are also multiplied, giving you a result of cm³.

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 r²

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

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 = s² = (50 m)² = 2,500 m²

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