Areas and Volumes in Similar Figures [H]
Higher Tier — This topic is assessed at Higher tier only (grades 4–9).
Areas and Volumes in Similar Figures
When two shapes are similar with linear scale factor \( k \), their areas and volumes are related by fixed multiples of \( k \). Understanding this three-tier relationship — length, area, volume — is essential for Higher tier problems involving similar solids.
The Three-Tier Relationship
| Quantity | Scale factor | Example: \( k=3 \) |
|---|---|---|
| Lengths | \( k \) | All lengths tripled |
| Areas | \( k^2 \) | Area multiplied by 9 |
| Volumes | \( k^3 \) | Volume multiplied by 27 |
Working Backwards
Given the area or volume ratio, find \( k \) by taking the appropriate root:
\[ k = \sqrt{\frac{A_2}{A_1}} \qquad k = \sqrt[3]{\frac{V_2}{V_1}} \]Then use \( k \) to find any corresponding length.
Mixed Problems
Some problems give a length ratio and ask for an area or volume, or give an area ratio and ask for a length or volume. The key is always to extract \( k \) from the given information, then apply the correct power.
Worked Examples
Two similar cones have base radii 4 cm and 10 cm. The smaller has volume 80 cm³. Find the volume of the larger.
\[ k = \frac{10}{4} = 2.5 \qquad V_{\text{large}} = 80\times2.5^3 = 80\times15.625 = 1{,}250 \text{ cm}^3 \]Two similar prisms have surface areas 48 cm² and 300 cm². The smaller has volume 64 cm³. Find the volume of the larger.
\[ k^2 = \frac{300}{48} = 6.25 \implies k = \sqrt{6.25} = 2.5 \] \[ V_{\text{large}} = 64\times2.5^3 = 64\times15.625 = 1{,}000 \text{ cm}^3 \]Two similar spheres have volumes in ratio 8 : 125. Find the ratio of their surface areas and the ratio of their radii.
\[ k^3 = \frac{125}{8} \implies k = \frac{5}{2} \] \[ \text{Radius ratio} = k = 5:2 \qquad \text{Surface area ratio} = k^2 = 25:4 \]Key Takeaways
- Linear scale factor \( k \): areas scale by \( k^2 \), volumes by \( k^3 \).
- From area ratio: \( k=\sqrt{A_2/A_1} \). From volume ratio: \( k=\sqrt[3]{V_2/V_1} \).
- Always find \( k \) first; then apply the correct power for the required quantity.
- The three-tier relationship applies to all similar shapes — 2D and 3D.