Discover Mathematics P6: Term 2, Unit 13, Lessons 3–4
GoalPredict how the volume changes when one, two or all three dimensions are doubled.
New words
double · to multiply by 2dimension · a length, a width or a heightratio V : 1 · the new volume compared with the old volume, which counts as 1
Explanation
Start with a cuboid 2 cm by 3 cm by 4 cm. Its volume is 2 × 3 × 4 = 24 cm³. If we double ONE dimension (say the length, 4 cm × 2), the volume is 4 × 3 × 4 = 48 cm³: it doubles (× 2). If we double TWO dimensions, the volume is 4 × 6 × 4 = 96 cm³: it is 4 times bigger (× 2 × 2). If we double ALL THREE, it is 4 × 6 × 8 = 192 cm³, which is 8 times bigger (× 2 × 2 × 2). Every doubled dimension multiplies the volume by 2. Watch out: doubling all the sides makes the surface area 4 times bigger but the volume 8 times bigger. If one dimension is doubled and another is halved, the volume stays the same.
Dimensions doubled
New size
Volume
V : 1
none
2 × 3 × 4
24
1
one
4 × 3 × 4
48
2
two
4 × 6 × 4
96
4
all three
4 × 6 × 8
192
8
Worked examples
The cuboid 2 × 3 × 4 has volume 24 cm³. Double the height: 2 × 3 × 8 = 48 cm³. The ratio of new to old is 48 : 24 = 2 : 1.
A cube of edge 2 cm has volume 8 cm³. Double the edge to 4 cm: volume 64 cm³. 64 ÷ 8 = 8, so the volume is 8 times bigger. The surface area of the cube grows only 4 times (24 to 96).
Predict first. A box is 6 cm by 4 cm by 5 cm (volume 120 cm³). Double the length and halve the width: 12 × 2 × 5 = 120 cm³. The volume does not change, because × 2 and × ½ cancel.
Class activity
Predict and test. Choose a cuboid with small whole numbers. Before you calculate, say what happens to the volume if you double two sides. Then calculate and check your prediction.
Practice
1
A cuboid is 3 cm by 2 cm by 5 cm. Find its volume, then the volume when the height is doubled.
