Volume of a box: learn and practice

Who this lesson is for

This volume lesson is for grades 7–8. Students multiply the three edges of a rectangular box. The practice link opens that skill at grade 7. Area of a rectangle, which uses only two edges, is a separate lesson for grades 4–5. Both grade hubs link to this one page.

Three edges, one product

Volume counts the unit cubes that fill a box. If the length is 2, the width is 3, and the height is 4, the box holds 2 × 3 × 4 = 24 cubes. You can see it as 4 layers, and each layer is a 2 by 3 rectangle of 6 cubes. Four layers make 24. The order of the three multiplications does not matter. Swapping length and height does not change how many cubes fit.

Volume is not the surface

The surface of the box is a wrapping. The volume is the space inside. A box can have a large volume and a different surface area. This page does not ask you to add the faces. If you add the three edges, 2 + 3 + 4 = 9, you have a length, not a volume. The unit is cubic: the answer 24 means 24 unit cubes.

Worked example 1: length 2, width 3, height 4

Find the volume of a box with length 2, width 3, and height 4. Multiply the three edges: 2 × 3 = 6, and 6 × 4 = 24. The bottom layer is 2 × 3 = 6 cubes. The height says there are 4 such layers, so 6 × 4 = 24. This is the catalog exercise. An answer of 9 adds the edges. An answer of 24 as a perimeter-style sum is the product, and that is the one that counts cubes.

Worked example 2: a flatter box

A box is 5 by 5 by 2. The base is a square layer of 5 × 5 = 25 cubes. Two layers make 25 × 2 = 50. So the volume is 50. You can also multiply 5 × 2 = 10 first, then 10 × 5 = 50. The same 50 appears. If the height were 1, the volume would equal the area of the base, 25. A height of 2 doubles that area into a volume of 50.

Worked example 3: a cube of side 3

A cube is a box whose three edges match. Side 3 means length 3, width 3, and height 3. The volume is 3 × 3 × 3 = 27. The first 3 × 3 = 9 is the area of one face, not the volume. You still have to multiply by the third edge. 27 unit cubes fill the cube. 3 × 3 = 9 would only pave one layer.

Checks before you leave the problem

Count three edges, not two. Multiply them. If you only have a face area, multiply by the remaining edge. The result counts cubes. Then practice box volume on the board.

Common questions about volume

Is volume length times width?

That product is the area of the base, one layer of cubes. Volume multiplies by the height as well. Three edges, not two.

Do the edges have to be multiplied in a special order?

No. 2 × 3 × 4, 4 × 2 × 3, and 3 × 4 × 2 are all 24. The picture of layers changes, and the number of cubes does not.

What unit should the answer use?

Cubic units, or a count of unit cubes. 24 for a 2 by 3 by 4 box means 24 cubes, not 24 units of length around an edge.