Volume of a Cuboid

Counting cubes

The volume of a solid is the amount of space it takes up. It is measured by counting the cubes that fill it, the way an area is measured by counting squares. Each cube is a unit cube: every edge of it is 1 unit long.

Here is a box 4 cubes long, 3 cubes high and 3 cubes deep, filled right up with unit cubes. Counting them one at a time is slow, and many of them are hidden inside the box where you cannot see them.

A box 4 cubes long, 3 high and 3 deep, filled with unit cubes.

One layer

Count one layer first. The front layer of the box is 4 cubes across and 3 cubes up. That is 3 rows of 4, an array, so it holds 4 × 3 = 12 cubes.

One layer on its own: 4 across and 3 up, 4 × 3 = 12 cubes.

Stack the layers

The box is 3 cubes deep, so it is 3 of those layers, one behind another. Every layer holds 12 cubes, so the whole box holds 12 × 3 = 36 cubes. Its volume is 36 cubes.

That is the rule for every cuboid. Multiply two edges to count the cubes in one layer, then multiply by the third edge to count the layers: volume = length × width × height. Any face can be the layer, so the order of the three edges does not matter: 4 × 3 × 3, 3 × 3 × 4 and 3 × 4 × 3 are all 36.

Three layers of 12 cubes: 12 × 3 = 36.

Every layer is the same 12 cubes as the base, so the count goes 12, 24, 36: the base times the number of layers.

Cubic units

A cube with edges 1 cm long has a volume of 1 cubic centimeter, written 1 cm³. A box 5 cm long, 4 cm wide and 3 cm high holds 5 × 4 = 20 of these cubes in its bottom layer, and it is 3 layers high, so its volume is 20 × 3 = 60 cm³.

The small 3 in cm³ is a reminder that three lengths were multiplied. All three must be in the same unit before you multiply. A length is measured in cm, an area in cm² and a volume in cm³, so an answer with the wrong unit has been worked out the wrong way.

Two ways to go wrong

Adding the three edges gives 4 + 3 + 3 = 10. That number counts no cubes at all: the box holds 36, far more than 10. Volume is found by multiplying, because every cube in a row is repeated in every row and every layer.

Stopping at 4 × 3 = 12 counts the front layer only. The box is 3 layers deep, so that 12 has to be counted 3 times.

Working backwards

Sometimes the volume is known and an edge is missing. A box holds 36 cubes, and its base is 4 cubes by 3 cubes. How deep is it?

One layer on that base holds 4 × 3 = 12 cubes. The question is how many layers of 12 make 36, and that is a division: 36 ÷ 12 = 3. The box is 3 layers deep. In general, the missing edge is the volume divided by the number of cubes in one layer.

36 cubes on a base of 4 by 3. Each layer is 12 cubes, so there are 36 ÷ 12 = 3 layers.

Water in a tank

The water in a rectangular tank is a cuboid too. Its length and width are the tank's, and its height is the depth of the water. A tank 20 cm long and 10 cm wide has a base of 20 × 10 = 200 square centimeters, so each centimeter of depth is a layer of 200 cm³ of water.

When a solid sinks to the bottom, it pushes the water up by its own volume. A stone of 1,000 cm³ makes the water rise by 1,000 ÷ 200 = 5 cm: the extra volume, divided by the layer on the base, counts how many centimeters of layers it makes.