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Similarity

9 min read

Same shape, different size

Two shapes are similar when they have the same shape but not necessarily the same size. Matching angles are equal and matching sides are in the same ratio. That ratio is the scale factor . A photograph and its enlargement are similar.

Similar trianglesTwo similar triangles: a small triangle with sides 3, 4, 5 and a larger triangle with sides 6, 8, 10. They have the same angles and every side of the larger one is twice the matching side of the smaller one, so the scale factor is 2.4358610scale factor 2: 6÷3 = 8÷4 = 10÷5 = 2the angles are equal
Every length is multiplied by the scale factor 2. The angles stay the same.
Lengths grow by k, areas by k², volumes by k³.
Try itEnlarge a 3-4-5 triangle

Change the scale factor . The sides grow by but the area grows by .

Worked exampleFind a missing side

Triangles ABC and DEF are similar. AB = 6 cm corresponds to DE = 9 cm. BC = 10 cm. Find EF.

  1. Scale factor from ABC to DEF.
  2. Multiply BC by k.

Answer: EF = 15 cm.

Worked exampleAreas of similar shapes

Two similar fields have lengths in the ratio 2 : 3. The smaller field has area 40 m². Find the area of the larger.

  1. Length ratio k = 3 ÷ 2 = 1.5.
  2. Area ratio is k².
  3. Multiply.

Answer: The larger field has area 90 m².

Check yourself A model car is a 1 : 20 scale of a real car. The real car is 4 m long. How long is the model?
Check yourself A shape is enlarged with scale factor 3. By what factor does its area grow?

Key terms

Similar
Same shape, sizes in proportion.
Scale factor
The number each length is multiplied by.

Summary

  • Similar shapes have equal angles and sides in the same ratio.
  • Find the scale factor k by dividing a new length by the matching old length.
  • Area scales by k² and volume by k³.
Congruence Geometrical Transformations