Math Detours · Grade 5 · Sharpen · 11

Counting cubes without counting

Volume is layers. Count one layer, multiply by the number of layers, and you have length × width × height.

Name ____________________Date ____________

Tonight's idea

Volume is how much space a box takes up. You measure it by counting the small cubes that fill the box.

4 long 3 wide one layer: 4 × 3 = 12 4 long 3 wide 2 high two layers: 12 × 2 = 24
One layer holds 12 cubes. The box is two layers.

This box is 4 cubes long, 3 cubes wide and 2 cubes high. You cannot count every cube in the picture, because some are hidden. Count in layers instead.

The bottom layer is a 4 by 3 rectangle of cubes: \(4 \times 3 = 12\). Every layer is the same, and there are 2 layers. So the box holds \(12 \times 2 = 24\) cubes.

Volume = cubes in one layer × number of layers = length × width × height.

A cube that is 1 cm along every edge is one cubic centimetre, written cm³. If these are centimetre cubes, the box has a volume of 24 cm³.

Watch one: a box 5 cm by 4 cm by 3 cm

One layer is 5 by 4, which is 20 cubes. There are 3 layers: \(20 \times 3 = 60\). The volume is 60 cm³.

Now stand the box on its end. A layer is 4 by 3, which is 12 cubes, and there are 5 layers. \(12 \times 5 = 60\) again.

It is the same box and the same cubes. So you can multiply the three numbers in any order.

Your turn

  1. A box is 6 cubes long, 2 cubes wide and 3 cubes high. (a) How many cubes are in the bottom layer? (b) How many cubes fill the box?
  2. Find the volume of each box in cm³. (a) 5 cm by 5 cm by 2 cm (b) 10 cm by 3 cm by 4 cm (c) a cube with every edge 3 cm
  3. (a) A box holds 40 cubes. Its bottom layer is 4 by 5. How high is the box? (b) Another box holds 60 cubes and is 3 cubes high. How many cubes are in one layer? Give two different rectangles the layer could be.
  4. You have 24 cubes and must use them all to build one solid box. A 4 by 3 by 2 box is one way. Find every other box you can build. A box turned on its side counts as the same box.

Stretch: the cubes nobody sees

A large cube is built from small cubes, 4 along every edge. It is dipped in paint, so every outside face is painted. Then it is taken apart. How many small cubes have no paint on them at all?

Answers

  1. (a) \(6 \times 2 = 12\). (b) \(12 \times 3 = 36\).
  2. (a) \(25 \times 2 = 50\) cm³. (b) \(30 \times 4 = 120\) cm³. (c) \(9 \times 3 = 27\) cm³.
  3. (a) One layer is 20 cubes, and \(40 \div 20 = 2\), so 2 high. (b) \(60 \div 3 = 20\) cubes. The layer could be 4 by 5, 2 by 10 or 1 by 20.
  4. Five more, six in all: 1 by 1 by 24, 1 by 2 by 12, 1 by 3 by 8, 1 by 4 by 6, 2 by 2 by 6, and 2 by 3 by 4. Each set of three numbers multiplies to 24.

Stretch. Take off the painted outside layer all round. Each edge loses one cube at both ends, so a 2 by 2 by 2 cube is left inside. \(2 \times 2 \times 2 = 8\) cubes have no paint. The other 56 have some.

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