Year 6 Maths reasoning (SATs): Similar shapes and scale factors
Part of the Year 6 Maths reasoning (SATs) ratio and proportion reasoning chapter. Chapter overview · whole chapter on one page
Similar shapes and scale factors
Rule
Two shapes are similar when one is an enlargement of the other. All lengths are multiplied by the same scale factor.
Scale factor = new length / original length. Missing length = original length x scale factor.
Key ideas
- Scale factor = new length / original length.
- Every length is multiplied by the same scale factor.
- Angles stay the same in similar shapes.
Method. Divide the new length by the original length.
- Write the scale factor as a division. The scale factor is the new length divided by the original.12 / 4
- Work it out. 12 / 4 = 3.3
- Write the answer. 3
Check. 4 x 3 = 12. ✓
Watch out. A scale factor bigger than 1 makes a shape bigger.
Method. Multiply each length of the original by the scale factor.
- Use the scale factor. Each length is multiplied by the scale factor 2.5.6 x 2.5
- Work it out. 6 x 2.5 = 15.15
- Write the answer. 15
Check. 6 x 2.5 = 15, and 8 x 2.5 = 20. ✓
Watch out. Check the scale factor with the side you know first.
Method. Divide the new length by the original length.
- Write the scale factor as a division. The scale factor is the new length divided by the original.20 / 5
- Work it out. 20 / 5 = 4.4
- Write the answer. 4
Check. 5 x 4 = 20. ✓
Watch out. A scale factor bigger than 1 makes a shape bigger.
Watch out
Watch out. Find the scale factor from one pair of matching sides, then use it for every other length.
Exam tip
Practice guide. Write the scale factor next to the shape, so you can use it again.
Exercise 3E Practice: similar shapes and scale factors
Non-calculatorMethod. Divide the new length by the original length.
- Write the scale factor as a division. The scale factor is the new length divided by the original.10 / 4
- Work it out. 10 / 4 = 2.5.2.5
- Write the answer. 2.5
Check. 4 x 2.5 = 10. ✓
Watch out. A scale factor bigger than 1 makes a shape bigger.
Method. Divide the new length by the original length.
- Write the scale factor as a division. The scale factor is the new length divided by the original.21 / 7
- Work it out. 21 / 7 = 3.3
- Write the answer. 3
Check. 7 x 3 = 21. ✓
Watch out. A scale factor bigger than 1 makes a shape bigger.
Method. Multiply each length of the original by the scale factor.
- Use the scale factor. Each length is multiplied by the scale factor 3.5 x 3
- Work it out. 5 x 3 = 15.15
- Write the answer. 15
Check. 5 x 3 = 15, and 4 x 3 = 12. ✓
Watch out. Check the scale factor with the side you know first.
Method. Multiply each length of the original by the scale factor.
- Use the scale factor. Each length is multiplied by the scale factor 2.7 x 2
- Work it out. 7 x 2 = 14.14
- Write the answer. 14
Check. 7 x 2 = 14, and 6 x 2 = 12. ✓
Watch out. Check the scale factor with the side you know first.
Method. Multiply each length of the original by the scale factor.
- Use the scale factor. Each length is multiplied by the scale factor 1.5.4 x 1.5
- Work it out. 4 x 1.5 = 6.6
- Write the answer. 6
Check. 4 x 1.5 = 6, and 10 x 1.5 = 15. ✓
Watch out. Check the scale factor with the side you know first.
Method. Find the scale factor from the widths, then use it for the height.
- Scale factor. 30 / 12 = 2.5.2.5
- New height. 8 x 2.5 = 20 cm.20 cm
- Write the answer. 20
Check. 12 x 2.5 = 30, and 8 x 2.5 = 20. ✓
Method. Multiply the length of the model by the scale factor 20.
- Scale factor. The scale 1:20 means each model length is multiplied by 20.20
- Real length. 15 x 20 = 300 cm.300 cm
- Write the answer. 300
Check. 300 / 20 = 15. ✓
Method. Test each choice against the question.
- Read what the question asks. Decide what the answer must do or show.
- Test the choice. 9 / 3 = 3, so the scale factor is 3.3 fits
- Reject the other choices. 6 would be 3 + 3, not a multiplication, and 1/3 would make the shape smaller.6 and 1/3 do not fit
- Write the answer. 3
Check. 3 is the one that fits. ✓
Method. Test each option.
- Check each option. Test each one against the rule or the calculation.
- Tick the true ones. In similar shapes the angles are equal and each length is multiplied by the same scale factor.All the angles stay the same, All lengths are multiplied by the same scale factor
- Leave the others unticked. These do not match.All the lengths stay the same, Corresponding lengths have different scale factors are not ticked
- Write the answer. All the angles stay the same, All lengths are multiplied by the same scale factor
Check. Each ticked option is true. ✓
Watch out. Lengths change in an enlargement, and different sides do not have different scale factors.
Method. Multiply the length by the scale factor.
- Multiply. 5 x 4 = 20.20
- Check. The new width is 3 x 4 = 12, so the shape is enlarged in both directions.20 cm long
- Write the answer. 20
Check. 20 / 4 = 5. ✓
Method. Use a square with side 1 cm.
- Start with the square. A 1 cm square has an area of 1 square cm.1 cm2
- Enlarge by 2. The new square has sides of 2 cm, so its area is 4 square cm.4 cm2
- Write the answer. No, because the area is multiplied by 2 x 2 = 4, not by 2.
Check. 4 = 2 x 2, not 2. ✓
Method. Find the scale factor from the matching sides, then multiply.
- Scale factor. 6 / 4 = 1.5.1.5
- Matching side. 10 x 1.5 = 15 cm.15 cm
- Write the answer. 15
Check. 4 x 1.5 = 6, so the scale factor is 1.5, and 10 x 1.5 = 15. ✓
Method. Find the scale factor for each pair of sides.
- Scale factor for the first pair. 6 / 2 = 3.3
- Scale factor for the second pair. 9 / 3 = 3.3
- Write the answer. Because 6 = 2 x 3 and 9 = 3 x 3, so both sides are multiplied by the same scale factor of 3.
Check. Both pairs have scale factor 3. ✓
Method. Multiply the plan length by the scale: each cm is 2 metres.
- Real length. 7 x 2 = 14 metres.14 metres
- Write the answer. 14
Check. 14 / 2 = 7. ✓