Year 6 Arithmetic (SATs): Similar shapes and scale factors
Part of the Year 6 Arithmetic (SATs) ratio and proportion chapter. Chapter overview · whole chapter on one page
Similar shapes and scale factors
Rule
In similar shapes, every length is multiplied by the same scale factor. Scale factor = new length ÷ old length.
A scale factor bigger than 1 makes the shape bigger; less than 1 makes it smaller.
Key ideas
- Scale factor = new length ÷ old length.
- Multiply every length by the same scale factor.
- Check the scale factor using a second pair of lengths.
Method. Find the scale factor, then multiply the height by it.
- Find the scale factor. The scale factor is the new side divided by the old side.14 ÷ 4 = 7/2
- Multiply the other side by the scale factor. Every length in the enlarged shape is multiplied by the same scale factor.6 × 7/2 = 21
- Write the answer. 21 cm
Check. 21 ÷ 6 = 3.5, and 14 ÷ 4 = 3.5. ✓
Method. The scale factor is 35 ÷ 5, and then multiply the other length.
- Find the scale factor. The scale factor is the new side divided by the old side.35 ÷ 5 = 7
- Multiply the other side by the scale factor. Every length in the enlarged shape is multiplied by the same scale factor.2 × 7 = 14
- Write the answer. 7 and 14 cm
Check. The scale factor is 7, and 2 × 7 = 14. ✓
Method. Find the scale factor, then multiply.
- Find the scale factor. The scale factor is the new side divided by the old side.12 ÷ 3 = 4
- Multiply the other side by the scale factor. Every length in the enlarged shape is multiplied by the same scale factor.5 × 4 = 20
- Write the answer. 20 cm
Check. 12 ÷ 3 = 4, and 5 × 4 = 20. ✓
Exercise 5D Practice: similar shapes and scale factors
Non-calculatorMethod. Find the scale factor, then multiply.
- Find the scale factor. The scale factor is the new side divided by the old side.35 ÷ 5 = 7
- Multiply the other side by the scale factor. Every length in the enlarged shape is multiplied by the same scale factor.4 × 7 = 28
- Write the answer. 28 cm
Check. The scale factor is 7, so 4 × 7 = 28. ✓
Method. Find the scale factor, then multiply.
- Find the scale factor. The scale factor is the new side divided by the old side.18 ÷ 3 = 6
- Multiply the other side by the scale factor. Every length in the enlarged shape is multiplied by the same scale factor.7 × 6 = 42
- Write the answer. 42 cm
Check. The scale factor is 6, so 7 × 6 = 42. ✓
Method. Find the scale factor, then multiply.
- Find the scale factor. The scale factor is the new side divided by the old side.8 ÷ 2 = 4
- Multiply the other side by the scale factor. Every length in the enlarged shape is multiplied by the same scale factor.5 × 4 = 20
- Write the answer. 20 cm
Check. The scale factor is 4, so 5 × 4 = 20. ✓
Method. Find the scale factor, then multiply.
- Find the scale factor. The scale factor is the new side divided by the old side.15 ÷ 6 = 5/2
- Multiply the other side by the scale factor. Every length in the enlarged shape is multiplied by the same scale factor.4 × 5/2 = 10
- Write the answer. 10 cm
Check. The scale factor is 5/2, so 4 × 5/2 = 10. ✓
Method. Find the scale factor, then multiply.
- Find the scale factor. The scale factor is the new side divided by the old side.12 ÷ 9 = 4/3
- Multiply the other side by the scale factor. Every length in the enlarged shape is multiplied by the same scale factor.6 × 4/3 = 8
- Write the answer. 8 cm
Check. The scale factor is 4/3, so 6 × 4/3 = 8. ✓
Method. Find the scale factor, then multiply.
- Find the scale factor. The scale factor is the new side divided by the old side.10 ÷ 4 = 5/2
- Multiply the other side by the scale factor. Every length in the enlarged shape is multiplied by the same scale factor.6 × 5/2 = 15
- Write the answer. 15 cm
Check. The scale factor is 5/2, so 6 × 5/2 = 15. ✓
Method. Divide the new length by the old length.
- Divide. 10 ÷ 4 = 2.5.2.5
- Write the answer. 2.5
Check. 4 × 2.5 = 10. ✓
Method. Find the scale factor, then the other side.
- Find the scale factor. The scale factor is the new side divided by the old side.12 ÷ 4 = 3
- Multiply the other side by the scale factor. Every length in the enlarged shape is multiplied by the same scale factor.9 × 3 = 27
- Write the answer. 27 cm
Check. 12 ÷ 4 = 3, and 9 × 3 = 27. ✓
Method. Find the scale factor, then multiply the height.
- Find the scale factor. The scale factor is the new side divided by the old side.9 ÷ 6 = 3/2
- Multiply the other side by the scale factor. Every length in the enlarged shape is multiplied by the same scale factor.10 × 3/2 = 15
- Write the answer. 15 cm
Check. 9 ÷ 6 = 1.5, and 10 × 1.5 = 15. ✓
Method. Divide the real length by the model length.
- Use centimetres. The real car is 300 cm, the model is 12 cm.300 cm and 12 cm
- Divide. 300 ÷ 12 = 25.25
- Write the answer. 25
Check. 12 × 25 = 300. ✓
Method. Check whether both lengths scale by the same factor.
- Width. 4 ÷ 2 = 2.2
- Height. 5 ÷ 3 = 5/3, which is not 2.5/3
- Write the answer. No, the scale factor for the width is 2 but for the height it is 5/3, so the shapes are not similar.
Check. 2 and 5/3 are different, so the shapes are not similar. ✓
Method. Divide 8 by 2, then check 20 ÷ 5 gives the same number.
- Find the scale factor. The scale factor is the new side divided by the old side.8 ÷ 2 = 4
- Multiply the other side by the scale factor. Every length in the enlarged shape is multiplied by the same scale factor.5 × 4 = 20
- Write the answer. 4
Check. 8 ÷ 2 = 4 and 20 ÷ 5 = 4, so the same scale factor is used. ✓