Similar shapes have equal angles, and their corresponding lengths are in the same ratio, the length scale factor k. Areas scale by k² and volumes by k³. So if every length doubles, the area becomes four times as big and the volume eight times as big. Given two areas or volumes, take the square root or cube root to recover the length scale factor.
Scale factors
- Length scale factor: k = new length ÷ original length
- Area scale factor: k²
- Volume scale factor: k³
- From areas: k = √(area ratio). From volumes: k = ∛(volume ratio)
Similar shapes and the length scale factor
Two shapes are similar if one is an enlargement of the other: the angles are equal and every pair of corresponding sides is in the same ratio. Find the length scale factor k by dividing a length on the larger shape by the matching length on the smaller one, then multiply other lengths by k. Matching the corresponding sides correctly is the key step.
Area scale factor
When lengths are multiplied by k, areas are multiplied by k², because area involves two lengths. With k = 3, an area of 14 cm² becomes 14 × 9 = 126 cm². This applies to any area, including surface area.
Volume scale factor
Volumes involve three lengths, so they are multiplied by k³. With k = 2, a volume of 45 cm³ becomes 45 × 8 = 360 cm³. Capacity and mass, for solids made of the same material, scale in the same way as volume.
Working backwards to the length scale factor
If you are given two areas, the length scale factor is the square root of their ratio: areas of 16 cm² and 100 cm² give k = √(100/16) = 2.5. From volumes, take the cube root. Once you have k, you can find any other length, area or volume.
Multi-step problems
Harder questions go from volumes to areas. If two similar solids have volumes 54 cm³ and 250 cm³, the volume ratio is 125/27, so k = 5/3. The area scale factor is then (5/3)² = 25/9, so a surface area of 36 cm² on the smaller solid becomes 100 cm² on the larger.
Worked examples
Example 1. Two triangles are similar. The smaller has base 4 cm and height 8 cm. The larger has base 10 cm. Find its height.
- k = 10 ÷ 4 = 2.5.
- Height = 8 × 2.5.
Answer: 20 cm.
Example 2. Two similar shapes have a length scale factor of 3. The smaller has an area of 14 cm². Find the area of the larger.
- Area scale factor: 3² = 9.
- 14 × 9.
Answer: 126 cm².
Example 3. Two similar solids have a length scale factor of 2. The smaller has a volume of 45 cm³. Find the volume of the larger.
- Volume scale factor: 2³ = 8.
- 45 × 8.
Answer: 360 cm³.
Example 4. Two similar shapes have areas 16 cm² and 100 cm². Find the length scale factor.
- Area ratio: 100 ÷ 16 = 6.25.
- k = √6.25.
Answer: 2.5.
Example 5. Two similar solids have volumes 54 cm³ and 250 cm³. The smaller has a surface area of 36 cm². Find the surface area of the larger.
- Volume ratio 250/54 = 125/27, so k = ∛(125/27) = 5/3.
- Area scale factor (5/3)² = 25/9.
- 36 × 25/9.
Answer: 100 cm².
Common mistakes
- Using k for areas. Areas scale by k², not k.
- Using k² for volumes. Volumes scale by k³.
- Matching the wrong sides. Identify corresponding sides from the equal angles.
- Square-rooting a volume ratio. For volumes, take the cube root to find k.
- Dividing the wrong way. k from small to large is large ÷ small.
How similar shapes are examined
Similar shapes are tested on both tiers of Edexcel GCSE (1MA1) and the International GCSE (4MA1); area and volume scale factors are Higher tier. Questions ask you to:
- find a missing length in similar triangles or shapes
- use the area scale factor to find an area
- use the volume scale factor to find a volume, capacity or mass
- work backwards from areas or volumes to a length
The similar shapes worksheet has worked answers, and the GCSE proportion revision engine provides more practice.
Similarity links to enlargement and volume and surface area.
Frequently asked questions
What makes two shapes similar?
Equal corresponding angles and corresponding sides in the same ratio.
If lengths double, what happens to the area?
It is multiplied by 4, because the area scale factor is 2².
How do you find the length scale factor from two volumes?
Divide the volumes and take the cube root.