Unit cubes and volume

A cubic unit fills space

A cube with edges 1 cm has volume 1 cm³, one cubic centimetre. It has six square faces, but six is not its volume. Volume measures the space filled; face area measures a surface.

Build a bottom layer with three cubes across and two cubes from front to back. That layer contains 3 × 2 = 6 cubes. Add an identical layer on top. There are 12 cubes altogether, so the solid’s volume is 12 cm³ when each cube is 1 cm³. Hidden cubes count too.

For a larger unit, a cube with edges 1 m is one cubic metre, 1 m³. State which unit cube is being counted. This lesson does not convert between cm³ and m³.

Read and make a drawing

On isometric dot paper, choose three grid directions for width, depth and height. Draw one cube with equal grid steps on its three edge directions. Build a 3-by-2 bottom layer, then mark a second layer. Use broken lines for hidden edges if they help you explain the structure.

A perspective drawing can hide cubes behind the front row. Say how many cubes are in each complete layer rather than counting only visible squares. A drawing should be checked against the arrangement; this page cannot certify the accuracy of a drawing made on paper.

Check your understanding 1 Some cubes cannot be seen

A solid has three complete layers. Each layer has two rows of four unit cubes. How many cubes are there? If one top corner cube is removed, what is the new volume in cubic units?

Your answer or drawing on paper

Show answer

Each layer has 2 × 4 = 8 cubes; three layers have 24. Removing one leaves 23 cubic units. It removes one cube of space, even though several visible faces change.

Same volume, different shape

Use 8 equal cubes to make a 2-by-2-by-2 cube, then rearrange them as one row of 8 cubes. Explain what changes and what stays the same.