# 10.7 Volumes

So far we have considered measuring lengths and areas, but most things in life are not flat; that is, two dimensional.

Questions like “How much does that hold?” need you to be able to specify the **volume** of an object. To do this, we can extend the ideas you learned earlier. To specify how much space something needs or occupies, you can count how many **cubes** of a certain size will fit into the space, in the same way that you counted how many square tiles covered an area. A cube is the same shape as a box, but all its sides are the same length and its six faces are all square.

Useful cubes to use are the cubic millimeter (written as mm^{3}), the cubic centimeter (written as cm^{3}) and the cubic meter (written as m^{3}).

In each case, the length of the side of the cube is 1 unit. So a cubic centimeter has all its sides of length 1 cm, as shown.

Now imagine filling a cubic centimeter with cubic millimeters. Since there are 10 mm in 1 cm, a cubic centimeter will contain 10 layers with each layer made up of 10 rows, each of 10 cubic millimeters.

So, . In other words, to convert a measurement in cubic centimeters into cubic millimeters, you would multiply by 1,000.

This makes sense—a cubic centimeter is larger than a cubic millimeter, so you would expect to need a lot more cubic millimeters to fill the same space.

## Activity: Converting Cubic Units

How many cubic centimeters are there in a cubic meter?

### Answer

There are 100 centimeters in a meter. Imagine a cubic meter—it will be 100 cm long, 100 cm wide, and 100 cm high. So you would be able to fit cubic centimeters into the cubic meter.

In other words, .

In Unit 5, you studied a different way of measuring volumes of liquids, for example, milliliters, centiliters, and liters. These units are linked to the cubic units because 1 ml has the same volume as , and 1,000 liters is the same as . The relationships between these different units are shown in the conversion diagram below. [ Can you explain how to get this diagram? ]

Suppose you have a box which measures 6 cm by 5 cm by 4 cm. What is its volume? Since all the dimensions are given in centimeters, you can measure the volume in cubic centimeters. Imagine filling the box with cubes of this size: six rows with five cubes in each row would cover the bottom of the box, and the box would be filled by four of these layers.

So, the total number of cubes used = . The volume of the box is therefore .

Before you move on, can you think of a suitable formula for working out the volume of a rectangular box?

Get some more practice with calculating volumes and with your mulitplication by playing this game [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)] .

10.6.8 Estimating Complicated Areas