Year 6: Similar shapes and scale factors
Part of the Year 6 ratio and proportion chapter. Chapter overview · whole chapter on one page
Similar shapes and scale factors
Two shapes are similar when one is an enlargement or a reduction of the other. Every length is multiplied by the same scale factor, and the angles stay the same. Look at the rectangle 3 cm by 5 cm. If it is enlarged by a scale factor of 4, the sides become 12 cm and 20 cm. The matching sides still make the same ratio: 3 : 12 is the same as 5 : 20.
To find a scale factor, divide a new length by its matching old length. To find a new length, multiply the old length by the scale factor. A scale factor between 0 and 1 makes the shape smaller.
Key ideas
- Similar shapes have the same angles and sides in the same ratio.
- Scale factor = new length ÷ old length, for a pair of matching sides.
- New length = old length × scale factor.
- A scale factor bigger than 1 enlarges the shape. A scale factor between 0 and 1 reduces it.
Key words. The scale factor is the number that every length is multiplied by. It is bigger than 1 for an enlargement and smaller than 1 for a reduction.
Method. Find the scale factor from one matching pair, then scale the other length.
- Find the scale factor: divide the new length by the old length. The scale factor multiplies every length.12 ÷ 3 = 4
- Multiply the other old length by the scale factor. Every length in a similar shape is scaled in the same way.5 × 4 = 20
- Write the answer. 20
Check. 20 ÷ 4 = 5, so the scale factor is right. ✓
Watch out. Use the scale factor on every length, not only on the side you were given.
Key words. The scale factor is the number that every length is multiplied by. It is bigger than 1 for an enlargement and smaller than 1 for a reduction.
Method. Find the scale factor from one matching pair, then scale the other length.
- Find the scale factor: divide the new length by the old length. The scale factor multiplies every length.10 ÷ 4 = 2.5
- Multiply the other old length by the scale factor. Every length in a similar shape is scaled in the same way.9 × 2.5 = 22.5
- Write the answer. 22.5
Check. 22.5 ÷ 2.5 = 9, so the scale factor is right. ✓
Watch out. Use the scale factor on every length, not only on the side you were given.
Key words. The scale factor is the number that every length is multiplied by. It is bigger than 1 for an enlargement and smaller than 1 for a reduction.
Method. Multiply the old length by the scale factor.
- Write the scale factor. It tells us what to multiply by.scale factor 0.25
- Multiply the old length by the scale factor. A scale factor below 1 makes the shape smaller.12 × 0.25 = 3
- Write the answer. 3
Check. 3 ÷ 0.25 = 12. ✓
Watch out. Multiply the length by the scale factor. Do not add it.
Watch out
Use the same scale factor for every length. A pupil who enlarges only the width of a rectangle does not have a similar shape.
Exam tip
Grown-up tip. Ask the child to find the scale factor first, then check it works for a second pair of sides. If the second pair gives a different number, the shapes are not similar.
Exercise 7D Similar shapes and scale factors
Non-calculatorKey words. The scale factor is the number that every length is multiplied by. It is bigger than 1 for an enlargement and smaller than 1 for a reduction.
Method. Multiply the old length by the scale factor.
- Write the scale factor. It tells us what to multiply by.scale factor 3
- Multiply the old length by the scale factor. A scale factor below 1 makes the shape smaller.5 × 3 = 15
- Write the answer. 15
Check. 15 ÷ 3 = 5. ✓
Watch out. Multiply the length by the scale factor. Do not add it.
Key words. The scale factor is the number that every length is multiplied by. It is bigger than 1 for an enlargement and smaller than 1 for a reduction.
Method. Multiply the old length by the scale factor.
- Write the scale factor. It tells us what to multiply by.scale factor 4
- Multiply the old length by the scale factor. A scale factor below 1 makes the shape smaller.2 × 4 = 8
- Write the answer. 8
Check. 8 ÷ 4 = 2. ✓
Watch out. Multiply the length by the scale factor. Do not add it.
Key words. The scale factor is the number that every length is multiplied by. It is bigger than 1 for an enlargement and smaller than 1 for a reduction.
Method. Multiply the old length by the scale factor.
- Write the scale factor. It tells us what to multiply by.scale factor 4
- Multiply the old length by the scale factor. A scale factor below 1 makes the shape smaller.7 × 4 = 28
- Write the answer. 28
Check. 28 ÷ 4 = 7. ✓
Watch out. Multiply the length by the scale factor. Do not add it.
Key words. The scale factor is the number that every length is multiplied by. It is bigger than 1 for an enlargement and smaller than 1 for a reduction.
Method. Multiply the old length by the scale factor.
- Write the scale factor. It tells us what to multiply by.scale factor 4
- Multiply the old length by the scale factor. A scale factor below 1 makes the shape smaller.3 × 4 = 12
- Write the answer. 12
Check. 12 ÷ 4 = 3. ✓
Watch out. Multiply the length by the scale factor. Do not add it.
Key words. The scale factor is the number that every length is multiplied by. It is bigger than 1 for an enlargement and smaller than 1 for a reduction.
Method. The scale factor is new length divided by old length.
- Write the old length and the new length that match. Match the sides that are the same.6 cm → 24 cm
- Divide the new length by the old length. This gives the scale factor.24 ÷ 6 = 4
- Write the answer. 4
Check. 6 × 4 = 24. ✓
Watch out. Divide the new length by the old length, not the other way round.
Key words. The scale factor is the number that every length is multiplied by. It is bigger than 1 for an enlargement and smaller than 1 for a reduction.
Method. The scale factor is new length divided by old length.
- Write the old length and the new length that match. Match the sides that are the same.9 cm → 3 cm
- Divide the new length by the old length. This gives the scale factor.3 ÷ 9 = 1/3
- Write the answer. 1/3
Check. 9 × 1/3 = 3. ✓
Watch out. Divide the new length by the old length, not the other way round.
Key words. The scale factor is the number that every length is multiplied by. It is bigger than 1 for an enlargement and smaller than 1 for a reduction.
Method. The scale factor is new length divided by old length.
- Write the old length and the new length that match. Match the sides that are the same.4 cm → 10 cm
- Divide the new length by the old length. This gives the scale factor.10 ÷ 4 = 2.5
- Write the answer. 2.5
Check. 4 × 2.5 = 10. ✓
Watch out. Divide the new length by the old length, not the other way round.
Key words. The scale factor is the number that every length is multiplied by. It is bigger than 1 for an enlargement and smaller than 1 for a reduction.
Method. The scale factor is new length divided by old length.
- Write the old length and the new length that match. Match the sides that are the same.6 cm → 3 cm
- Divide the new length by the old length. This gives the scale factor.3 ÷ 6 = 0.5
- Write the answer. 0.5
Check. 6 × 0.5 = 3. ✓
Watch out. Divide the new length by the old length, not the other way round.
Key words. The scale factor is the number that every length is multiplied by. It is bigger than 1 for an enlargement and smaller than 1 for a reduction.
Method. Find the scale factor from one matching pair, then scale the other length.
- Find the scale factor: divide the new length by the old length. The scale factor multiplies every length.45 ÷ 15 = 3
- Multiply the other old length by the scale factor. Every length in a similar shape is scaled in the same way.10 × 3 = 30
- Write the answer. 30
Check. 30 ÷ 3 = 10, so the scale factor is right. ✓
Watch out. Use the scale factor on every length, not only on the side you were given.
Key words. The scale factor is the number that every length is multiplied by. It is bigger than 1 for an enlargement and smaller than 1 for a reduction.
Method. The scale factor is new length divided by old length.
- Write the old length and the new length that match. Match the sides that are the same.6 cm → 450 cm
- Divide the new length by the old length. This gives the scale factor.450 ÷ 6 = 75
- Write the answer. 75
Check. 6 × 75 = 450. ✓
Watch out. Divide the new length by the old length, not the other way round.
Key words. The scale factor is the number that every length is multiplied by. It is bigger than 1 for an enlargement and smaller than 1 for a reduction.
Method. Find the scale factor from one matching pair, then scale the other length.
- Find the scale factor: divide the new length by the old length. The scale factor multiplies every length.12 ÷ 4 = 3
- Multiply the other old length by the scale factor. Every length in a similar shape is scaled in the same way.9 × 3 = 27
- Write the answer. 27
Check. 27 ÷ 3 = 9, so the scale factor is right. ✓
Watch out. Use the scale factor on every length, not only on the side you were given.
Key words. The scale factor is the number that every length is multiplied by. It is bigger than 1 for an enlargement and smaller than 1 for a reduction.
Method. Multiply the old length by the scale factor.
- Write the scale factor. It tells us what to multiply by.scale factor 2.5
- Multiply the old length by the scale factor. A scale factor below 1 makes the shape smaller.8 × 2.5 = 20
- Write the answer. 20
Check. 20 ÷ 2.5 = 8. ✓
Watch out. Multiply the length by the scale factor. Do not add it.
Key words. The scale factor is the number that every length is multiplied by. It is bigger than 1 for an enlargement and smaller than 1 for a reduction.
Method. Multiply the old length by the scale factor.
- Write the scale factor. It tells us what to multiply by.scale factor 2.5
- Multiply the old length by the scale factor. A scale factor below 1 makes the shape smaller.5 × 2.5 = 12.5
- Write the answer. 12.5
Check. 12.5 ÷ 2.5 = 5. ✓
Watch out. Multiply the length by the scale factor. Do not add it.
Key words. The scale factor is the number that every length is multiplied by. It is bigger than 1 for an enlargement and smaller than 1 for a reduction.
Method. Find the scale factor for the width and for the height. If they match, the shapes are similar.
- Find the scale factor for the width: the new width divided by the old width. Matching sides give the scale factor.6 ÷ 2 = 3
- Find the scale factor for the height: the new height divided by the old height. The shapes are similar only if this is the same.15 ÷ 5 = 3
- Compare the two scale factors. They are both 3, so the sides are in the same ratio.3 = 3
- Write the answer. Yes. Both lengths are multiplied by 3.
Check. 2 × 3 = 6 and 5 × 3 = 15. ✓
Watch out. Check both sides, not just one.
Key words. The scale factor is the number that every length is multiplied by. It is bigger than 1 for an enlargement and smaller than 1 for a reduction.
Method. Find the scale factor for the width and for the height. If they are different, the shapes are not similar.
- Find the scale factor for the width: the new width divided by the old width. Matching sides give the scale factor.4 ÷ 3 = 4/3
- Find the scale factor for the height: the new height divided by the old height. The shapes are similar only if this is the same.10 ÷ 8 = 1.25
- Compare the two scale factors. They are different, so the sides do not grow in the same way.4/3 ≠ 1.25
- Write the answer. No. The scale factors are different: 4/3 and 5/4.
Check. Using 3 and 4 gives 4/3, but 8 and 10 gives 5/4. ✓
Watch out. The scale factors must be equal for the shapes to be similar.
Key words. The scale factor is the number that every length is multiplied by. It is bigger than 1 for an enlargement and smaller than 1 for a reduction.
Method. Scale both lengths first, then multiply them to get the new area.
- Multiply each length by the scale factor. Both sides are scaled.5 × 4 = 20 cm and 3 × 4 = 12 cm
- Find the area of the new rectangle. Area is length times width.20 × 12 = 240
- Write the answer. The area is in square centimetres.240
Check. The old area is 5 × 3 = 15, and 15 × 16 = 240. ✓
Watch out. Area changes by the scale factor times itself, not by the scale factor alone.
Key words. The scale factor is the number that every length is multiplied by. It is bigger than 1 for an enlargement and smaller than 1 for a reduction.
Method. The area is scaled by the scale factor multiplied by itself.
- Write the scale factor. Every length is multiplied by this.scale factor 4
- Multiply the scale factor by itself, because area is length times width. Both lengths are scaled.4 × 4 = 16
- Write the answer. 16
Check. Check with the rectangle: 4 × width and 4 × length gives 16 times the area. ✓
Watch out. A scale factor of 4 makes the area 16 times bigger, not 4 times bigger.