1. Overview
Similarity in geometry means that two shapes have the same angles and their sides are in proportion, even if they are different sizes. This allows you to calculate unknown lengths, areas, or volumes by using scale factors. Similarity is used in maps, models, and many real-world applications.
Key Definitions
- Similar Shapes: Two shapes are similar if their corresponding angles are equal and their corresponding sides are in the same ratio.
- Scale Factor ($k$): The ratio of any two corresponding lengths in similar figures.
- Corresponding Sides: Sides that are in the same relative position in two similar shapes.
- Congruent: Shapes that are identical in both shape and size (a scale factor of 1).
Core Content
Calculating Lengths of Similar Shapes
To find missing lengths in similar shapes, you must first determine the Linear Scale Factor ($k$).
The Method:
- Identify a pair of corresponding sides where both lengths are known.
- Calculate the scale factor: $k = \text{Length on Shape B} \div \text{Length on Shape A}$.
- To find a missing length on the larger shape, multiply the corresponding length by $k$.
- To find a missing length on the smaller shape, divide the corresponding length by $k$.
Worked example 1 — Finding a side length
Question: Triangle ABC and triangle DEF are similar. AB = 8 cm, DE = 12 cm, and BC = 6 cm. Find the length of EF.
Step 1: Identify corresponding sides. AB and DE are corresponding sides. BC and EF are corresponding sides.
Step 2: Calculate the Scale Factor ($k$). $k = \frac{DE}{AB}$ $k = \frac{12}{8}$ $k = 1.5$
Step 3: Find the missing side EF. $EF = BC \times k$ $EF = 6 \times 1.5$ $EF = 9\text{ cm}$
Answer: EF = 9 cm
Worked example 2 — Finding a side length (smaller shape)
Question: Two similar rectangles PQRS and WXYZ are given. PQ = 15 cm, WX = 5 cm, and YZ = 3 cm. Find the length of RS.
Step 1: Identify corresponding sides. PQ and WX are corresponding sides. RS and YZ are corresponding sides.
Step 2: Calculate the Scale Factor ($k$). $k = \frac{WX}{PQ}$ $k = \frac{5}{15}$ $k = \frac{1}{3}$
Step 3: Find the missing side RS. $RS = YZ \div k$ $RS = 3 \div \frac{1}{3}$ $RS = 3 \times 3$ $RS = 9\text{ cm}$
Answer: RS = 9 cm
Calculator Tip: In non-calculator papers, scale factors are usually simple integers (like 2, 3, or 5) or simple fractions (like 1.5). In calculator papers, you may get decimals; keep the full value in your calculator to avoid rounding errors mid-calculation.
Extended Content (Extended curriculum only)
Area and Volume Relationships
When shapes are similar, the ratios of their areas and volumes are not the same as the ratio of their lengths. They follow a power rule based on the linear scale factor ($k$).
- Length Scale Factor $= k$
- Area Scale Factor $= k^2$
- Volume Scale Factor $= k^3$
Worked Example 1: Area Two similar rectangles have lengths of 3 cm and 6 cm respectively. If the area of the smaller rectangle is 10 cm², find the area of the larger rectangle.
Step 1: Find the linear scale factor ($k$). $k = 6 \div 3$ $k = 2$
Step 2: Find the area scale factor ($k^2$). $\text{Area Scale Factor} = 2^2$ $\text{Area Scale Factor} = 4$
Step 3: Calculate the new area. $\text{Area} = 10 \times 4$ $\text{Area} = 40\text{ cm}^2$
Answer: The area of the larger rectangle is 40 cm²
Worked Example 2: Volume (Working Backwards) Two similar solids have volumes of $16\text{ cm}^3$ and $54\text{ cm}^3$. The height of the smaller solid is 4 cm. Find the height of the larger solid.
Step 1: Find the Volume Scale Factor. $\text{Volume Scale Factor} = \frac{54}{16}$ $\text{Volume Scale Factor} = 3.375$
Step 2: Find the Linear Scale Factor ($k$) by taking the cube root. $k = \sqrt[3]{3.375}$ $k = 1.5$
Step 3: Calculate the missing height. $\text{Height} = 4 \times 1.5$ $\text{Height} = 6\text{ cm}$
Answer: The height of the larger solid is 6 cm
Worked example 3 — Finding the area of a smaller shape
Question: Two similar pentagons have corresponding sides of 4 cm and 12 cm. The area of the larger pentagon is 36 cm². Calculate the area of the smaller pentagon.
Step 1: Calculate the linear scale factor ($k$). $k = \frac{4}{12}$ $k = \frac{1}{3}$
Step 2: Calculate the area scale factor ($k^2$). $\text{Area Scale Factor} = (\frac{1}{3})^2$ $\text{Area Scale Factor} = \frac{1}{9}$
Step 3: Calculate the area of the smaller pentagon. $\text{Area of smaller pentagon} = 36 \times \frac{1}{9}$ $\text{Area of smaller pentagon} = 4\text{ cm}^2$
Answer: The area of the smaller pentagon is 4 cm²
Worked example 4 — Finding the volume of a larger shape
Question: Two similar spheres have radii of 2 cm and 5 cm. The volume of the smaller sphere is 33.5 cm³. Calculate the volume of the larger sphere.
Step 1: Calculate the linear scale factor ($k$). $k = \frac{5}{2}$ $k = 2.5$
Step 2: Calculate the volume scale factor ($k^3$). $\text{Volume Scale Factor} = (2.5)^3$ $\text{Volume Scale Factor} = 15.625$
Step 3: Calculate the volume of the larger sphere. $\text{Volume of larger sphere} = 33.5 \times 15.625$ $\text{Volume of larger sphere} = 523.125\text{ cm}^3$
Answer: The volume of the larger sphere is 523.125 cm³
Key Equations
Note: These formulas are not provided on the IGCSE formula sheet; they must be memorised.
Linear Ratio: $k = \frac{L_2}{L_1}$ where $L$ = length
Area Ratio: $\frac{A_2}{A_1} = k^2$ where $A$ = area/surface area
Volume Ratio: $\frac{V_2}{V_1} = k^3$ where $V$ = volume
Common Mistakes to Avoid
- ❌ Wrong: Using the linear scale factor to find area (e.g., if length doubles, saying area doubles).
- ✓ Right: Always square the linear scale factor for area (if length doubles, area increases by $2^2 = 4$).
- ❌ Wrong: Using the linear scale factor to find volume (e.g., if length doubles, saying volume doubles).
- ✓ Right: Always cube the linear scale factor for volume (if length doubles, volume increases by $2^3 = 8$).
- ❌ Wrong: Writing 'SAS' (Side-Angle-Side) for a triangle congruence proof when you have used the hypotenuse and one other side of a right-angled triangle.
- ✓ Right: Use 'RHS' (Right angle-Hypotenuse-Side) for congruence proofs involving right-angled triangles.
- ❌ Wrong: Assuming two triangles are similar just because they look alike, without proving it.
- ✓ Right: You must prove similarity by showing two angles are equal (AA) or all three sides are in proportion (SSS).
Exam Tips
- Parallel Lines: If an exam question shows a small triangle inside a larger triangle with a common vertex and parallel bases, the triangles are similar.
- Command Words: "Show that" means you must write down every step of your logic, especially the calculation of the scale factor.
- Contexts: Similarity often appears in questions involving shadows (triangles formed by the sun), maps (scale drawings), or liquid being poured into conical containers.
- Check Units: Ensure all lengths are in the same units (e.g., all cm) before calculating the scale factor. If a volume is in litres and a length is in cm, convert 1 litre to $1000\text{ cm}^3$ first.