Use square and cube roots to recover dimensions

What you will learn

• Use square and cube roots to recover dimensions

Models used: number sentences, stories, diagrams.

Before you start

• Relate volume base area and height

Recover the missing layers

The cuboid contains 36 cubes. Its base is 4 by 3, so each layer contains 12.

1. Count cubes across a layer, then count the layers. Some cubes are hidden.

   The cuboid contains 36 cubes. Its base is 4 by 3, so each layer contains 12.

   4 × 3 = 12 cubes per layer

   Width 4, height 1, depth 3

2. Count cubes across a layer, then count the layers. Some cubes are hidden.

   Two layers contain 24 cubes. We need one more layer to reach 36.

   Width 4, height 2, depth 3

3. Count cubes across a layer, then count the layers. Some cubes are hidden.

   There are three layers, so the height is 3 units.

   height = 36 ÷ 12 = 3

   Width 4, height 3, depth 3

Worked example: A new example, together

A cube has volume 27 cm³. Find its edge length.

1. Work out the known face first, then divide the volume by it.

   Roots undo repeated multiplication of the same edge length.

2. Equal edges multiply three times to give volume.

   A cube has volume 27 cm³. Find its edge length.

   3 cm

3. Use the cube root of 27.

4. 3 × 3 × 3 = 27.

5. Check the result against the question we started with.

   3 × 3 × 3 = 27.

   We found 3 cm. Roots undo repeated multiplication of the same edge length.

Common mistakes to discuss

• Dividing by one edge only. Volume is the product of three lengths. With volume 60 cm³ on a 5 cm by 4 cm base, the height is 60 ÷ 20 = 3 cm.

Check your understanding

• Use square and cube roots to recover dimensions
• work back from a volume to a missing dimension
• check the answer against the original quantities and units

Explain your reasoning to an adult, then try a fresh question in the interactive lesson. A result from one attempt is not proof of lasting mastery.