Similarity

Maths · Geometry. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

Two shapes are similar when one is an enlargement of the other. Their matching angles are equal and their matching sides are all in the same ratio. This ratio is the scale factor.

Two triangles are similar if two pairs of angles are equal. This happens when a line is drawn parallel to one side of a triangle, and when two straight lines cross between parallel lines. Match the sides that are opposite equal angles.

Areas and volumes do not grow at the same rate as lengths. If every length is multiplied by k, every area is multiplied by k² and every volume by k³.

Rules to remember

Length scale factor k = length on the larger shape ÷ matching length on the smaller shape
Missing length on the larger shape = matching length on the smaller shape × k
Area scale factor = k² (this includes surface area)
Volume scale factor = k³ (this also works for mass and capacity when the material is the same)
Going back: k = √(area ratio) = cube root of (volume ratio)
Line DE parallel to BC in triangle ABC: triangles ADE and ABC are similar, so AD/AB = AE/AC = DE/BC

Common mistake

In a triangle with a parallel line, students use AD/DB as the scale factor. The sides of the big triangle are the whole lengths, so use AD/AB, where AB = AD + DB.

Worked example

Two similar jugs have heights 10 cm and 25 cm. The smaller jug holds 0.4 litres. How much does the larger jug hold?

Length scale factor k = 25 ÷ 10 = 2.5.
Volume scale factor = k³ = 2.5 × 2.5 × 2.5 = 15.625.
Larger volume = 0.4 × 15.625 = 6.25.

Answer: 6.25 litres

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