Learning Objectives
  • Use linear scale factors to calculate corresponding lengths.
  • Use squared scale factors for areas and surface areas.
  • Use cubed scale factors for volumes.
  • Work backwards from area or volume ratios to find a linear scale factor.
  • Solve compound similarity problems with units and proportional reasoning.
Key Terms
Similar figures
Figures with equal corresponding angles and proportional corresponding lengths.
Linear scale factor
The ratio of corresponding lengths.
Area scale factor
The square of the linear scale factor.
Volume scale factor
The cube of the linear scale factor.
Corresponding
Occupying the same relative position in similar figures.
Surface area
The total area of all outer faces or curved surfaces of a solid.
What This Chapter Covers
  • Lengths in similar figures
  • Area and surface-area ratios
  • Volume ratios
  • Reverse calculations using roots
  • Units and multi-stage similarity problems
Similarity And Scale Factor

Similar figures have the same shape but may have different sizes. Corresponding angles are equal and corresponding lengths are in a constant ratio. The first step is to match corresponding sides correctly.

k=\frac{\text{image length}}{\text{original length}}
\frac{A_2}{A_1}=k^2
\frac{V_2}{V_1}=k^3
Lengths
Worked Example: Corresponding Length

Question: Two similar triangles have scale factor \frac{5}{3} from small to large. A side of the small triangle is 9 cm. Find the corresponding large side.

  1. Multiply by the linear scale factor.
  2. 9\times\frac53=15.

Answer: 15 cm.

Areas And Surface Areas

Area is two-dimensional, so it scales with the square of the length factor. Surface area also has square units and uses the same squared factor.

Worked Example: Area Scale

Question: Two similar shapes have corresponding lengths 4 cm and 10 cm. The smaller area is 24 cm². Find the larger area.

  1. Linear factor =10/4=2.5.
  2. Area factor =2.5^2=6.25.
  3. Larger area =24\times6.25.

Answer: 150 cm².

Volumes

Volume is three-dimensional, so it scales with the cube of the length factor.

Worked Example: Volume Scale

Question: Two similar solids have linear scale factor 3 from small to large. The smaller volume is 40 cm³. Find the larger volume.

  1. Volume factor =3^3=27.
  2. Larger volume =40\times27.

Answer: 1080 cm³.

Working Backwards

To recover a linear factor from an area ratio, take a square root. From a volume ratio, take a cube root.

k=\sqrt{\frac{A_2}{A_1}}\qquad\text{or}\qquad k=\sqrt[3]{\frac{V_2}{V_1}}
Worked Example: From Area Ratio

Question: The areas of two similar figures are in ratio 49:81. Find the corresponding length ratio.

  1. Take square roots of both parts.
  2. \sqrt{49}:\sqrt{81}=7:9.

Answer: Length ratio 7:9.

Worked Example: From Volume Ratio

Question: The volumes of two similar solids are in ratio 64:125. Find their surface-area ratio.

  1. Cube roots give linear ratio 4:5.
  2. Square the linear ratio for surface area.

Answer: Surface-area ratio 16:25.

Units And Direction

Use the scale factor in the correct direction. If working from large to small, the factor may be less than 1. Convert units before comparing corresponding lengths. Area ratios do not carry units, but calculated areas must use square units and volumes cube units.

Examination Guidance
  • Match corresponding sides before forming a ratio.
  • Write the direction of the scale factor: small to large or large to small.
  • Square for area and surface area; cube for volume.
  • Take roots when working backwards.
  • Check units: length, square units and cubic units must not be mixed.
Common Mistakes
  • Using the linear factor directly for area or volume.
  • Squaring when a volume factor is required.
  • Using the scale factor in the reverse direction.
  • Comparing non-corresponding sides.
  • Forgetting to convert units before forming a ratio.
Knowledge Check And Practice

1. A linear scale factor is 4. What is the area factor?

Answer: 16.

2. A linear scale factor is 4. What is the volume factor?

Answer: 64.

3. Areas are in ratio 36:49. Find the length ratio.

Answer: 6:7.

4. Volumes are in ratio 27:8. Find the length ratio.

Answer: 3:2.

5. Length ratio is 2:5. Find surface-area ratio.

Answer: 4:25.

6. Length ratio is 3:4. Find volume ratio.

Answer: 27:64.
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