Similarity

Similar shapes have the same shape but can be different sizes: their angles are equal and all their lengths are in the same ratio, the scale factor. Areas grow by the square of the scale factor and volumes by its cube.

Key facts

  • Similar shapes: length scale factor k = new length ÷ old length.
  • Area scale factor = k².
  • Volume scale factor = k³.
  • Equal angles (from parallel lines or vertically opposite angles) often show that triangles are similar.

Worked examples

Example 1

The triangles are similar. Find x.

  1. Match the longest sides: 6 cm matches 12 cm. k = 12 ÷ 6 = 2
  2. Check with another pair: 4 cm matches 8 cm, also × 2.
  3. x matches the 5 cm side. x = 5 × 2 = 10 cm

Answer: x = 10 cm

Example 2

Two triangles are similar with a scale factor of 3. A side of the small triangle is 4 cm. How long is the matching side of the large one?

  1. Multiply by the scale factor. 4 × 3 = 12 cm

Answer: 12 cm

Example 3

Are triangles with sides 3, 4, 5 and 6, 8, 10 similar?

  1. Divide each side of the big one by the matching small side. 6 ÷ 3 = 2, 8 ÷ 4 = 2, 10 ÷ 5 = 2
  2. All the ratios are the same, so they are similar.

Answer: Yes (scale factor 2)

Example 4

A photo 10 cm wide and 15 cm tall is enlarged to 25 cm wide. How tall is it?

  1. Find the scale factor from the widths. 25 ÷ 10 = 2.5
  2. Multiply the height by it. 15 × 2.5 = 37.5 cm

Answer: 37.5 cm

Example 5

Two similar shapes have lengths in the ratio 2 : 5. The smaller has an area of 12 cm². Find the area of the larger.

  1. Picture squares with sides 2 and 5. They hold 4 and 25 unit squares. 2² = 4, 5² = 25
  2. Areas grow by the length factor squared. (5/2)² = 25/4 = 6.25
  3. Larger area: 12 × 6.25 = 75 cm²

Answer: 75 cm²

Example 6

Two similar jugs hold 250 ml and 2,000 ml. The small jug is 10 cm tall. How tall is the large jug?

  1. Volume scale factor: 2,000 ÷ 250 = 8
  2. Volumes scale by k³, so take the cube root to get the length factor. ∛8 = 2
  3. Height: 10 × 2 = 20 cm

Answer: 20 cm

Example 7

Two similar cylinders have volumes 16 cm³ and 54 cm³. The small one is 4 cm tall. How tall is the large one?

  1. Divide the volumes. 54 ÷ 16 = 27/8
  2. Take the cube root for the length scale factor. ∛(27/8) = 3/2
  3. Multiply the height. 4 × 3/2 = 6 cm

Answer: 6 cm

Example 8

Two similar triangles have areas of 9 cm² and 25 cm². A side of the smaller one is 6 cm. Find the matching side of the larger one.

  1. The area scale factor is the square of the length scale factor. 25/9
  2. Take the square root. √(25/9) = 5/3
  3. Multiply. 6 × 5/3 = 10 cm

Answer: 10 cm