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MathidiaMathematics Academy
Similarity and congruence
Lesson 42 of 75
Easy practice

Similar lengths

Objective. Use linear scale factors to calculate missing lengths.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Similar shapes have the same proportions but different sizes, meaning corresponding angles are equal and corresponding sides are all in the same ratio, called the linear scale factor. To find a missing length in a similar shape, divide a known pair of corresponding sides to find the scale factor, then multiply or divide the length you need to find by that factor. For example, if a small triangle has a side of 4 cm matching a 10 cm side on a larger similar triangle, the scale factor is 10 ÷ 4 = 2.5, so any other side on the small triangle is multiplied by 2.5 to find its match. Always check that sides are matched correctly between the two shapes before calculating.

Worked example

Two similar rectangles have corresponding lengths of 6 cm and 15 cm. If the smaller rectangle has a width of 4 cm, find the width of the larger rectangle.

Diagram of a rectangle
  1. 1Find the scale factor by dividing the larger length by the smaller length: 15 ÷ 6.
  2. 215 ÷ 6 = 2.5, so the scale factor is 2.5.
  3. 3Multiply the smaller rectangle's width by this scale factor: 4 × 2.5.
  4. 44 × 2.5 = 10.

Answer: 10 cm

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Common mistake. Pupils sometimes divide the sides the wrong way round, calculating the small shape's side divided by the large shape's side when scaling up, producing a scale factor less than 1 by mistake. Always divide the target shape's side by the known matching side.

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