Perimeter, area and volume – Higher only
When we enlarge a shape by a scale factor, the length of each edge and the perimeter are multiplied by the scale factor.
When we enlarge a shape by a scale factor, the area of the shape is multiplied by the square of the scale factor.
When we enlarge a shape by a scale factor, the volume of the shape is multiplied by the cube of the scale factor.
Question
Two similar posters have areas of 24 cm2 and 384 cm2. If the smaller poster has a perimeter of 20 cm, calculate the perimeter of the larger poster.
Scale factor for area = new area ÷ old area = 384 ÷ 24 = 16
Area is the scale factor squared, so we need to find the square root of 16.
\(\sqrt{16} = {4}\)
Scale factor for length = 4
Perimeter of larger poster = 20 × 4 = 80 cm.
Question
Two similar tanks are filled with water. One has a capacity of 30 m3, the other a capacity of 240 cm3. Calculate the scale factor for the lengths of the tanks and the scale factor for the surface areas.
Scale factor for volume = 240 ÷ 30 = 8
Volume is the scale factor cubed, so we need to find the cube root of 8.
\(\sqrt[3]{8}~=~{2}\)
Scale factor for length = 2
Area is the scale factor squared:
22 = 2 × 2 = 4
Scale factor for area = 4
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