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 mm3), the cubic centimeter (written as cm3) and the cubic meter (written as m3).

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, equation sequence one times cm super three equals 10 multiplication 10 multiplication 10 times mm super three equals one comma 000 times mm super three. 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 Symbol Activity: Converting Cubic Units

How many cubic centimeters are there in a cubic meter?

Solution Symbol

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 100 multiplication 100 multiplication 100 cubic centimeters into the cubic meter.

In other words, equation sequence one times m super three equals 100 multiplication 100 multiplication 100 times cm super three equals one comma 000 comma 000 times cm super three.

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 one times cm super three, and 1,000 liters is the same as one times m super three. 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 = six multiplication five multiplication four equals 120. The volume of the box is therefore 120 times cm super three.

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

Videoclip iconGet 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

10.7.1 Volume Formulas