Area, volume and surface area
Calculating the volume of cubes and cuboids
Split a cuboid into centimetre cubes and count them. A 4 by 3 by 2 cuboid has 12 cubes in each layer and two layers — 24 in total. That count is the volume.
Cubic units
A cube 1 cm long, 1 cm wide and 1 cm high is a CENTIMETRE CUBE. Its VOLUME is 1 CUBIC CENTIMETRE (1 cm³). Volume is how much space a solid takes up, so its units are cubed — cm³, mm³, m³. Lengths use cm, areas use cm², volumes use cm³. The little number tells you how many dimensions are involved.
Counting the cubes
Take a cuboid 4 cm long, 3 cm wide and 2 cm high, and split it into centimetre cubes. Each layer is 4 × 3 = 12 cubes. There are 2 layers. 12 × 2 = 24 cubes So the volume is 24 cm³. The formula is just a shortcut for that count.
The formula
Volume = length × width × height V = l × w × h For a cuboid 8 cm by 5 cm by 3 cm: V = 8 × 5 × 3 = 120 cm³ The three lengths can be multiplied in any order. And if all the lengths are in metres, the answer is in m³.
Compound solids
A compound solid splits into a cube and a cuboid. Do each, then add. Cube of side 4 m: V = 4 × 4 × 4 = 64 m³ Cuboid 9 m by 4 m by 7 m: V = 9 × 4 × 7 = 252 m³ Total: 64 + 252 = 316 m³ A cube is just a cuboid whose three lengths happen to be equal.
Worked examples
A cuboid 8 cm by 5 cm by 3 cm
- V = l × w × h.
- 8 × 5 = 40.
- 40 × 3 = 120.
Answer: 120 cm³
A cuboid 4 cm by 3 cm by 2 cm, by counting
- Each layer is 4 × 3 = 12 cubes.
- There are 2 layers.
- 12 × 2 = 24.
Answer: 24 cm³
A cube of side 4 m
- All three lengths are 4.
- 4 × 4 × 4 = 64.
Answer: 64 m³
A compound solid: a 4 m cube plus a 9 by 4 by 7 cuboid
- Cube: 4 × 4 × 4 = 64 m³.
- Cuboid: 9 × 4 × 7 = 252 m³.
- 64 + 252 = 316.
Answer: 316 m³
A cuboid of volume 60 cm³, 5 cm long and 4 cm wide
- 60 = 5 × 4 × h, so 60 = 20h.
- 60 ÷ 20 = 3.
Answer: 3 cm high
Why cm³ and not cm²?
- A volume has three dimensions, not two.
- So the unit is cubed.
Answer: Because volume uses length, width AND height
Where you see this in real life
- Shipping is charged by volume as well as weight, so a box's three dimensions decide the price.
- Concrete is ordered in cubic metres, worked out from the length, width and depth of what you are filling.
Key words
- Volume
- the amount of space a solid takes up
- Cubic centimetre
- the volume of a 1 cm cube, written cm³
- Cuboid
- a box shape with six rectangular faces
- Cube
- a cuboid whose length, width and height are equal
- Dimensions
- the length, width and height of a solid