Ratio in Similarity

Ratio in Similarity

Two shapes are similar if one is an enlargement of the other — all corresponding angles are equal and all corresponding lengths are in the same ratio (the scale factor). Similarity connects ratio notation, scale factors, and the properties of lengths, areas and volumes directly to real geometric problems.

Finding the Scale Factor

The linear scale factor \( k \) is found by dividing any corresponding length in the larger shape by the corresponding length in the smaller:

\[ k = \frac{\text{image length}}{\text{original length}} \]

All other corresponding lengths are also in the ratio \( 1 : k \). The scale factor works in both directions: \( k \) to go from small to large; \( 1/k \) to go from large to small.

Areas and Volumes of Similar Shapes

If the linear scale factor is \( k \):

  • Area scale factor: \( k^2 \)
  • Volume scale factor: \( k^3 \)

These are the same laws met in B1678, now applied to prove similarity and to solve missing-length problems in similar figures.

Trigonometric Ratios as Scale Factors

In a right-angled triangle, the trigonometric ratios (sin, cos, tan) are themselves ratios of lengths. They are constant for a given angle because all right-angled triangles with the same acute angle are similar — the sides are always in the same proportion regardless of size.

\[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \qquad \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \qquad \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \]

Worked Examples

Two similar triangles have corresponding sides 6 cm and 15 cm. Find the scale factor. A third side of the smaller triangle is 8 cm — find the corresponding side of the larger.
\[ k = 15 \div 6 = 2.5 \qquad \text{larger side} = 8 \times 2.5 = 20 \text{ cm} \]
Two similar cylinders have heights 4 cm and 10 cm. The smaller has surface area 48 cm² and volume 32 cm³. Find the surface area and volume of the larger.
\[ k = 10 \div 4 = 2.5 \qquad \text{Area} = 48 \times 2.5^2 = 48 \times 6.25 = 300 \text{ cm}^2 \] \[ \text{Volume} = 32 \times 2.5^3 = 32 \times 15.625 = 500 \text{ cm}^3 \]
Explain why \( \sin 30° \) is always \( 0.5 \), regardless of the size of the right-angled triangle.

All right-angled triangles with a 30° angle are similar to each other — one is an enlargement of another. The ratio opposite ÷ hypotenuse is therefore the same for all of them, regardless of size. This constant ratio is defined as \( \sin 30° = 0.5 \).

 Key Takeaways

  • Similar shapes: all corresponding lengths in ratio \( 1 : k \). Find \( k \) from any pair of known corresponding lengths.
  • Area scales as \( k^2 \); volume scales as \( k^3 \). To find \( k \) from areas: \( k = \sqrt{\text{area ratio}} \). From volumes: \( k = \sqrt[3]{\text{volume ratio}} \).
  • Trigonometric ratios are constant ratios of side lengths — they are the scale factors of similar right-angled triangles.
  • If given two corresponding areas, find \( k \) by taking the square root before working with lengths.