Similarity · Year 11
In Year 11 you learn: area and volume in similar shapes.
Similar shapes have the same shape but can be different sizes: their angles are equal and all their lengths are in the same ratio, the scale factor. Areas grow by the square of the scale factor and volumes by its cube.
Key facts
- Similar shapes: length scale factor k = new length ÷ old length.
- Area scale factor = k².
- Volume scale factor = k³.
- Equal angles (from parallel lines or vertically opposite angles) often show that triangles are similar.
Worked examples
Example 1
The triangles are similar. Find x.
Answer: x = 10 cm
Example 2
Two triangles are similar with a scale factor of 3. A side of the small triangle is 4 cm. How long is the matching side of the large one?
Answer: 12 cm
Example 3
Are triangles with sides 3, 4, 5 and 6, 8, 10 similar?
Answer: Yes (scale factor 2)
Example 4
A photo 10 cm wide and 15 cm tall is enlarged to 25 cm wide. How tall is it?
Answer: 37.5 cm
Example 5
Two similar shapes have lengths in the ratio 2 : 5. The smaller has an area of 12 cm². Find the area of the larger.
Answer: 75 cm²
Example 6
Two similar jugs hold 250 ml and 2,000 ml. The small jug is 10 cm tall. How tall is the large jug?
Answer: 20 cm
Example 7
Two similar cylinders have volumes 16 cm³ and 54 cm³. The small one is 4 cm tall. How tall is the large one?
Answer: 6 cm
Example 8
Two similar triangles have areas of 9 cm² and 25 cm². A side of the smaller one is 6 cm. Find the matching side of the larger one.
Answer: 10 cm