Volume of Rectangular Prisms with Unit Cubes | Grade 7 Math | LevelKite

Volume is the amount of space inside a solid. To measure it, you use a unit cube, which is a cube whose edges are each 1 unit long. A cube with 1 cm edges has a volume of 1 cubic centimetre ( $1\text{ cm}^3$ ). A cube with 1 in edges has a volume of 1 cubic inch.

In this lesson, every rectangular prism has positive whole-number edge lengths in one unit and is packed with unit cubes of that unit, with no gaps or overlaps. Its volume is the number of unit cubes that pack it. Edge lengths are in linear units, such as centimetres. Volume is in cubic units of the same unit, such as cubic centimetres.

Take a rectangular prism that is 4 cm long, 3 cm high and 2 cm wide from front to back. It is packed with centimetre cubes. Each cube is 1 cm on a side, so:

4 cubes run along the 4 cm length,
3 cubes run along the 3 cm height,
2 cubes run along the 2 cm width.

You can slice the packed box into front-to-back layers. Each of these layers is a wall of cubes one cube thick that faces you. The figure below shows one such layer as its own box. It has 4 cubes across and 3 cubes high, so it holds $4 \times 3 = 12$ cubes.

One front-to-back layer:

4

cubes across (4 cm) and

3

high (3 cm), so

4 \times 3 = 12

cubes.

The full box: two front-to-back layers of

12

cubes, so

12 + 12 = 24

cubic centimetres.

The full box has 2 front-to-back layers because its width is 2 cm. Its front face draws every cube of the front layer. Each strip on the top and right faces is one whole layer. You cannot see the cubes in the back layer, but they are there because the box is packed with no gaps.

Add the layers one at a time: $12 + 12 = 24$ . This is the same as $12 \times 2 = 24$ , which is the cubes in one layer times the number of layers. The volume is $24\text{ cm}^3$ .

You can also slice the same box into bottom layers. Recall that a base of a rectangular prism is either face of a pair of opposite faces, and a bottom layer is one layer of cubes resting on the base.

The box sits on a base that is 4 cm long and 2 cm wide, so one bottom layer holds $4 \times 2 = 8$ cubes. The height of 3 cm is measured straight up from the base, at a right angle to it. That means 3 bottom layers stack up: $8 + 8 + 8 = 24$ .

Writing $(4 \times 2) \times 3$ instead of $(4 \times 3) \times 2$ changes the order of the 3 and the 2, so a different pair is multiplied first. Multiplication gives the same product in any order and grouping, so the total is still 24 cubes.

The figure below is the same box turned so its bottom faces you. The length runs across, the width runs up, and the height runs from front to back. The front layer you see is now one bottom layer of the box.

The same box turned: across is the length (4 cm), up is the width (2 cm), front to back is the height (3 cm). Bottom layer

4 \times 2 = 8

cubes,

3

layers:

24

cubic centimetres.

Key concept

Volume as layers of unit cubes

When a rectangular prism with whole-number edge lengths in one unit is packed with unit cubes with no gaps or overlaps, its volume is the number of cubes inside it.

To find it, count the cubes in one layer and match each edge to the cubes along it. Then add the layers one at a time, or multiply by the number of layers. Front-to-back, bottom and side layers all give the same total. Write the volume in cubic units of the cube's unit.

\text{volume} = (\text{cubes in one layer}) \times (\text{number of layers})

Worked example

A small box packed with inch cubes

A small box is 3 in long, 2 in high and 3 in wide from front to back. It is packed with inch cubes with no gaps or overlaps. Find its volume.

Match each edge to its cubes. Each cube is 1 in on a side, so 3 cubes run along the 3 in length, 2 along the 2 in height, and 3 along the 3 in width.
Count one front-to-back layer. It is 3 cubes across and 2 cubes high, so multiply the cubes across by the cubes high.
$3 \times 2 = 6$
The 3 in width gives 3 front-to-back layers. Add them one at a time, because each layer holds the same 6 cubes.
$6 + 6 + 6 = 18$
Check with bottom layers. One bottom layer has length times width cubes. The 2 in height gives 2 bottom layers. Using the width before the height changes the order of the factors, but not the product.
$3 \times 3 = 9,\quad 9 + 9 = 18$

Answer

The small box has a volume of $18$ cubic inches.

The small box:

3

inch cubes across (length),

2

high (height),

3

layers front to back (width), so

6 + 6 + 6 = 18

cubic inches.

Worked example

A crate counted in words

A shipping crate is 6 ft long, 4 ft wide and 5 ft high. It is packed with 1-foot cubes with no gaps or overlaps. Find its volume.

Match the edges to the cubes: 6 cubes along the length, 4 along the width and 5 along the height.
One front-to-back layer is 6 cubes across and 5 cubes high.
$6 \times 5 = 30$
The 4 ft width gives 4 layers. Add them one at a time.
$30 + 30 = 60,\; 60 + 30 = 90,\; 90 + 30 = 120$
Check with bottom layers: $6 \times 4 = 24$ cubes in each, and 5 layers for the 5 ft height. Swapping the order of the 5 and the 4 does not change the product.
$24 + 24 + 24 + 24 + 24 = 120$

Answer

The crate has a volume of $120$ cubic feet.

Takeaway: The volume of a rectangular prism packed with unit cubes is the number of cubes inside it, including the hidden ones. Count the cubes in one layer, then add the layers one at a time, or multiply by the number of layers. Any direction of layers gives the same total. The answer is in cubic units of the cube's unit.