Skip to content

Volume by counting cubes

Measurement · level 4, lesson 5 of 7

A question to start

A small box is 2 cm long, 2 cm wide and 2 cm tall. You build a big box that is twice as long, twice as wide and twice as tall: 4 cm by 4 cm by 4 cm. How many of the small boxes would fit inside the big one?

In level 3 you saw that doubling the sides of a rug gives four times the area. For boxes the answer is bigger still: eight small boxes fit inside the big one. By the end of this lesson you will see exactly why, and you will be able to find how much space is inside any box.

The idea

What volume is

Volume is the amount of space a solid object takes up, or the amount of space inside a hollow one. A brick, a cereal box, a room and a planet all have a volume.

Length is measured with little lines (units of length). Area is measured with little squares. Volume is measured with little cubes. A cube is a solid with six square faces, all the same size, like a dice.

  • A cubic centimetre (cm³) is a cube 1 cm long, 1 cm wide and 1 cm tall. The tiny "ones" cubes used in many classrooms for place value are exactly this size.
  • A cubic metre (m³) is a cube 1 m on every edge. Twelve metre sticks taped into a cube frame would show one; a grown-up could crouch inside it.

The little raised 3 in cm³ is read "cubed". It is a reminder that a cube has three directions: length, width and height.

Filling a box, layer by layer

The simplest solid to measure is a rectangular box (mathematicians call it a rectangular prism or a cuboid). Here is a box 4 cm long, 3 cm wide and 2 cm tall, built from centimetre cubes.

length 42width 3 one layer: 4 × 3 = 122 layers: 2 × 12 = 24

Count it in layers, the way a builder lays bricks.

  1. One layer. The bottom layer is a rectangle of cubes, 4 long and 3 wide. From level 3 you know a rectangle of 3 rows of 4 has \(3 \times 4 = 12\) squares, so the bottom layer has 12 cubes. Notice that 12 is also the area of the bottom of the box, in cm².
  2. How many layers. The box is 2 cm tall, so there are 2 layers, each exactly like the first.
  3. Total. \(2 \times 12 = 24\) cubes. The volume is 24 cm³.

The same steps work for every box. The length and width decide how many cubes are in a layer: length × width. The height decides how many layers: height. So

\[\text{volume of a box} = \text{length} \times \text{width} \times \text{height}, \qquad V = l \times w \times h\]

or, in the layer way of saying it,

\[V = \text{area of the base} \times \text{height}\]

Keep the second form in mind. In level 5 it gives the volume of every prism and cylinder, because they too are stacks of identical layers.

The order does not matter

You could count the same box in upright slices instead: 4 slices from left to right, each one 3 deep and 2 tall (\(3 \times 2 = 6\) cubes), giving \(4 \times 6 = 24\). Or front to back: 3 slices, each \(4 \times 2 = 8\) cubes, giving \(3 \times 8 = 24\). Every way gives 24, because it is the same box of cubes. That is why you can multiply the three lengths in any order.

Cubes and "cubed"

A cube has length, width and height all equal. A cube with edges of 5 cm has volume \(5 \times 5 \times 5 = 125\) cm³. A number times itself three times is called the number "cubed", written \(5^3\), because it is the volume of a cube. The cube numbers 1, 8, 27, 64, 125, ... are the volumes of cubes with edges 1, 2, 3, 4, 5, ...

Why doubling the edges gives 8 times the volume

1 cubeedges doubled: 2 × 2 × 2 = 8 cubes

Doubling the length puts 2 cubes in each row. Doubling the width makes 2 rows in each layer, so each layer has \(2 \times 2 = 4\) cubes. Doubling the height makes 2 layers: \(4 \times 2 = 8\) cubes. Each doubling multiplies by 2, and there are three directions, so the volume is multiplied by \(2 \times 2 \times 2 = 8\).

For lengths: double. For areas (two directions): four times. For volumes (three directions): eight times. Make every edge 3 times as long and the volume becomes \(3 \times 3 \times 3 = 27\) times as big. This is why a pot twice as wide and twice as tall holds so much more soup than you might guess, and why Mars, a bit more than half as wide as Earth, has only roughly a seventh of Earth's volume (lesson 7).

Parts of a cube

Edges are not always whole numbers. A box 4 cm by 3 cm by 2.5 cm has 2 full layers of 12 cubes and one half-layer on top, which is like 6 more cubes: \(24 + 6 = 30\) cm³. The formula agrees: \(4 \times 3 \times 2.5 = 30\) cm³. Just as in level 3 for area, the formula works for parts of units too.

Same unit, every time

All three lengths must be in the same unit before you multiply, and then the volume is in that unit cubed. A box 2 m by 50 cm by 1 m is \(2 \times 0.5 \times 1 = 1\) m³ (not \(2 \times 50 \times 1 = 100\) of anything).

Worked examples

1. A cereal box. A box is 20 cm long, 8 cm wide and 30 cm tall. Its volume?

Base: \(20 \times 8 = 160\) cm². Layers: 30. Volume: \(160 \times 30 = 4800\) cm³.

2. A cube. A cube has edges of 6 cm. Its volume is \(6 \times 6 \times 6 = 216\) cm³.

3. Missing height. A box has volume 360 cm³. Its base is 10 cm by 6 cm. How tall is it?

Each layer has \(10 \times 6 = 60\) cubes. \(360 \div 60 = 6\) layers, so the box is 6 cm tall.

4. Packing. How many 2 cm cubes fit in a box 10 cm by 6 cm by 4 cm?

Along the length, \(10 \div 2 = 5\) cubes fit; across, \(6 \div 2 = 3\); up, \(4 \div 2 = 2\). So \(5 \times 3 \times 2 = 30\) cubes. (Check: the box is \(10 \times 6 \times 4 = 240\) cm³, each 2 cm cube is \(2 \times 2 \times 2 = 8\) cm³, and \(240 \div 8 = 30\). ✓)

5. A room. A bedroom is 4 m long, 3 m wide and 2.5 m high. How many cubic metres of air does it hold?

\(4 \times 3 \times 2.5 = 30\) m³.

6. A staircase of cubes. Steps are built from centimetre cubes: the bottom step is a row of 3 cubes, the next a row of 2 on top, then 1 on top of that, and the whole staircase is 4 cubes deep (front to back). How many cubes?

Look at the side: \(3 + 2 + 1 = 6\) squares. It is 4 cubes deep, so it is 4 slices of 6: \(4 \times 6 = 24\) cm³. "Area of the side × depth" works here too, because every slice front to back is the same shape.

Try it

  1. Find the volume: (a) a box 5 cm by 4 cm by 3 cm; (b) a cube with edges 10 cm; (c) a box 2.5 m by 2 m by 4 m.
  2. A box has a base of area 48 cm² and is 7 cm tall. What is its volume?
  3. A box has volume 200 cm³, length 10 cm and width 5 cm. How tall is it?
  4. How many 1 cm cubes fit in a cube with edges of 4 cm? How many 2 cm cubes?
  5. A cube has volume 64 cm³. How long is each edge?
  6. A sandbox is 2 m long and 1.5 m wide. It is filled with sand 0.4 m deep. How many cubic metres of sand?
  7. Box A is 6 cm by 4 cm by 2 cm. Box B is 3 cm by 4 cm by 4 cm. Which holds more?
  8. Every edge of a box is made 3 times as long. How many times bigger is its volume? Check with a 1 by 2 by 3 box.
Answers
  1. (a) \(5 \times 4 \times 3 = 60\) cm³. (b) \(10 \times 10 \times 10 = 1000\) cm³. (c) \(2.5 \times 2 \times 4 = 20\) m³.
  2. \(48 \times 7 = 336\) cm³.
  3. One layer has \(10 \times 5 = 50\) cm³, and \(200 \div 50 = 4\) layers, so the box is 4 cm tall.
  4. \(4 \times 4 \times 4 = 64\) centimetre cubes. 2 cm cubes: 2 fit along each edge, so \(2 \times 2 \times 2 = 8\).
  5. Which number cubed is 64? \(4 \times 4 \times 4 = 64\), so each edge is 4 cm.
  6. \(2 \times 1.5 \times 0.4 = 1.2\) m³.
  7. A: \(6 \times 4 \times 2 = 48\) cm³. B: \(3 \times 4 \times 4 = 48\) cm³. They hold the same, though they look different.
  8. \(3 \times 3 \times 3 = 27\) times. Check: the 1 by 2 by 3 box has volume 6; the 3 by 6 by 9 box has volume \(3 \times 6 \times 9 = 162\), and \(162 \div 6 = 27\). ✓

Common mistake

Adding the edges, or multiplying only two. \(4 + 3 + 2 = 9\) is not the volume of a 4 by 3 by 2 box, and \(4 \times 3 = 12\) is only one layer. Volume needs all three lengths, multiplied.

Writing the wrong unit. Volume is in cubic units (cm³, m³), area in square units (cm², m²), length in plain units (cm, m). A box's volume written as "24 cm" or "24 cm²" is a different kind of measurement altogether.

Where it leads