Higher only · area and volume scale factors

Similar shapes.
Lengths × k, areas × k²,
volumes × k³.

the most common
lost mark in similarity

enlarge a shape by scale factor k —

k · k² · k³

length has 1 dimension, area 2, volume 3

so k is used once, twice or three times

step 1 of 4

Why area is × k²

1. Enlarge a 1 cm square by scale factor 3.

2. It is 3 wide and 3 tall: 9 squares fit.

✓ Lengths × 3, so area × 3² = 9

step 2 of 4

Area

1. Length scale factor from matching sides.

2. Square it for the area scale factor.

✓ Multiply the small area by k²

lengths 4 cm and 10 cm

k = 10 ÷ 4 = 2.5

k² = 6.25

area 12 × 6.25 = 75 cm²

step 3 of 4

Volume

1. Cube the length scale factor.

2. Multiply the small volume by it.

✓ Going to the smaller shape? Divide

k = 2.5

k³ = 2.5³ = 15.625

volume 8 × 15.625 = 125 cm³

step 4 of 4

Working backwards

1. Dividing the areas gives k², not k.

2. Square root to get the length factor.

✓ Always get back to k first

areas 20 cm² and 45 cm²

k² = 45 ÷ 20 = 2.25

k = √2.25 = 1.5

volumes × 1.5³ = × 3.375

similar shapes —

Length

× k

Area

× k²

Volume

× k³

Given areas or volumes?
Root back to k first.

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