GCSE Maths: Area and volume scale factors (Foundation)
Part of the GCSE Maths transformations and similarity chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Area and volume scale factors
When a shape is enlarged by a scale factor k, every length is multiplied by k. What happens to the area and the volume? Look at squares. A square with side 1 has area 1. If the side is doubled to 2, the square holds 2 × 2 = 4 of the little squares, so the area is multiplied by 4 (not 2). If the side is trebled to 3, the area is 3 × 3 = 9: multiplied by 9.
For cubes it is the same idea in three directions. A cube with side 2 is made of 2 × 2 × 2 = 8 little cubes, so doubling the lengths multiplies the volume by 8. Trebling the lengths gives 3 × 3 × 3 = 27 little cubes.
This works for any similar shapes, not only squares and cubes. If the length scale factor is k:
| Quantity | Scale factor |
|---|---|
| Lengths (sides, perimeters, heights, radii) | k |
| Areas (surface area, cross-section area) | k2 |
| Volumes (also capacity and, for the same material, mass) | k3 |
Using ratio notation. If the lengths of two similar shapes are in the ratio a : b, then their areas are in the ratio a2 : b2 and their volumes in the ratio a3 : b3. For example, if two similar vases have heights in the ratio 2 : 3, their surface areas are in the ratio 4 : 9 and their volumes in the ratio 8 : 27.
Key ideas
- If the length scale factor is k, the area scale factor is k2 and the volume scale factor is k3.
- Length ratio a : b gives area ratio a2 : b2 and volume ratio a3 : b3.
- Higher: to find k from areas take the square root; from volumes take the cube root.
- This works for similar shapes only.
- Surface areas use k2 and capacities use k3.
Key words. The length scale factor k says how many times longer every length becomes. The area scale factor is k2 and the volume scale factor is k3. These work only for similar shapes or solids.
Method. For similar shapes, if the lengths are multiplied by k, the areas are multiplied by k2.
- Write down the length scale factor. It is the number that every length is multiplied by.k = 3
- Work out the area scale factor. Area has two lengths in it (length × width), so it is multiplied by k twice: k2.area scale factor = 32 = 3 × 3 = 9
- Multiply the original area by the area scale factor. The new value is the old value multiplied by the scale factor.12 × 9 = 108 cm2
Check. Divide back: 108 ÷ 9 = 12. ✓
Watch out. Do not multiply the area or volume by k. Area uses k2, volume uses k3.
Key words. The length scale factor k says how many times longer every length becomes. The area scale factor is k2 and the volume scale factor is k3. These work only for similar shapes or solids.
Method. For similar shapes, if the lengths are multiplied by k, the volumes are multiplied by k3.
- Write down the length scale factor. It is the number that every length is multiplied by.k = 2
- Work out the volume scale factor. Volume has three lengths in it (length × width × height), so it is multiplied by k three times: k3.volume scale factor = 23 = 2 × 2 × 2 = 8
- Multiply the original volume by the volume scale factor. The new value is the old value multiplied by the scale factor.15 × 8 = 120 cm3
Check. Divide back: 120 ÷ 8 = 15. ✓
Watch out. Do not multiply the area or volume by k. Area uses k2, volume uses k3.
Key words. The length scale factor k says how many times longer every length becomes. The area scale factor is k2 and the volume scale factor is k3. These work only for similar shapes or solids.
Method. If the lengths of two similar shapes are in the ratio a : b, then the areas are in the ratio a2 : b2 and the volumes are in the ratio a3 : b3.
- Write the length ratio in simplest form. Divide both numbers by their highest common factor.Lengths: 2 : 5
- Square both numbers for the area ratio. The area scale factor is the square of the length scale factor.Areas: 22 : 52 = 4 : 25
- Cube both numbers for the volume ratio. The volume scale factor is the cube of the length scale factor.Volumes: 23 : 53 = 8 : 125
Check. Test with a = 2, b = 5: a cube of side 2 has volume 8 and a cube of side 5 has volume 125. ✓
Watch out. Square (or cube) each number in the ratio, not just one of them.
Watch out
Do not multiply the area or the volume by the length scale factor. If the lengths double, the area goes up by a factor of 4 and the volume by a factor of 8. And when you are given an area ratio, take the square root to get the length ratio, not half of it.Exercise 21J Area and volume scale factors
Non-calculatorKey words. The length scale factor k says how many times longer every length becomes. The area scale factor is k2 and the volume scale factor is k3. These work only for similar shapes or solids.
Method. For similar shapes, if the lengths are multiplied by k, the areas are multiplied by k2.
- Write down the length scale factor. It is the number that every length is multiplied by.k = 2
- Work out the area scale factor. Area has two lengths in it (length × width), so it is multiplied by k twice: k2.area scale factor = 22 = 2 × 2 = 4
- Multiply the original area by the area scale factor. The new value is the old value multiplied by the scale factor.5 × 4 = 20 cm2
Check. Divide back: 20 ÷ 4 = 5. ✓
Watch out. Do not multiply the area or volume by k. Area uses k2, volume uses k3.
Key words. The length scale factor k says how many times longer every length becomes. The area scale factor is k2 and the volume scale factor is k3. These work only for similar shapes or solids.
Method. For similar shapes, if the lengths are multiplied by k, the areas are multiplied by k2.
- Write down the length scale factor. It is the number that every length is multiplied by.k = 3
- Work out the area scale factor. Area has two lengths in it (length × width), so it is multiplied by k twice: k2.area scale factor = 32 = 3 × 3 = 9
- Multiply the original area by the area scale factor. The new value is the old value multiplied by the scale factor.7 × 9 = 63 cm2
Check. Divide back: 63 ÷ 9 = 7. ✓
Watch out. Do not multiply the area or volume by k. Area uses k2, volume uses k3.
Key words. The length scale factor k says how many times longer every length becomes. The area scale factor is k2 and the volume scale factor is k3. These work only for similar shapes or solids.
Method. For similar shapes, if the lengths are multiplied by k, the volumes are multiplied by k3.
- Write down the length scale factor. It is the number that every length is multiplied by.k = 2
- Work out the volume scale factor. Volume has three lengths in it (length × width × height), so it is multiplied by k three times: k3.volume scale factor = 23 = 2 × 2 × 2 = 8
- Multiply the original volume by the volume scale factor. The new value is the old value multiplied by the scale factor.9 × 8 = 72 cm3
Check. Divide back: 72 ÷ 8 = 9. ✓
Watch out. Do not multiply the area or volume by k. Area uses k2, volume uses k3.
Key words. The length scale factor k says how many times longer every length becomes. The area scale factor is k2 and the volume scale factor is k3. These work only for similar shapes or solids.
Method. For similar shapes, if the lengths are multiplied by k, the volumes are multiplied by k3.
- Write down the length scale factor. It is the number that every length is multiplied by.k = 3
- Work out the volume scale factor. Volume has three lengths in it (length × width × height), so it is multiplied by k three times: k3.volume scale factor = 33 = 3 × 3 × 3 = 27
- Multiply the original volume by the volume scale factor. The new value is the old value multiplied by the scale factor.4 × 27 = 108 cm3
Check. Divide back: 108 ÷ 27 = 4. ✓
Watch out. Do not multiply the area or volume by k. Area uses k2, volume uses k3.
Key words. The length scale factor k says how many times longer every length becomes. The area scale factor is k2 and the volume scale factor is k3. These work only for similar shapes or solids.
Method. For similar shapes, if the lengths are multiplied by k, the areas are multiplied by k2.
- Write down the length scale factor. It is the number that every length is multiplied by.k = 4
- Work out the area scale factor. Area has two lengths in it (length × width), so it is multiplied by k twice: k2.area scale factor = 42 = 4 × 4 = 16
- Multiply the original area by the area scale factor. The new value is the old value multiplied by the scale factor.3 × 16 = 48 cm2
Check. Divide back: 48 ÷ 16 = 3. ✓
Watch out. Do not multiply the area or volume by k. Area uses k2, volume uses k3.
Method. Area of a square = side × side.
- Smaller square. 3 × 3 = 9 cm2
- Larger square. 6 × 6 = 36 cm2
Check. 36 ÷ 9 = 4 and the sides doubled (6 ÷ 3 = 2): 22 = 4. ✓
Watch out. Square the side, do not double it.
Method. Write the two areas as a ratio and simplify.
- Write the ratio. Smaller : larger.9 : 36
- Simplify. Divide both by 9.9 ÷ 9 = 1 and 36 ÷ 9 = 4, so 1 : 4
Check. 1 × 9 = 9 and 4 × 9 = 36. ✓
Watch out. Divide both numbers by the highest common factor.
Method. Compare the two ratios.
- Ratio of the lengths. 3 : 6 simplifies by dividing by 3.1 : 2
- Compare with the areas. The area ratio 1 : 4 is the length ratio squared.12 : 22 = 1 : 4
Check. This agrees with the rule: area ratio = (length ratio)2. ✓
Watch out. The area ratio is the square of the length ratio.
Key words. The length scale factor k says how many times longer every length becomes. The area scale factor is k2 and the volume scale factor is k3. These work only for similar shapes or solids.
Method. For similar shapes, if the lengths are multiplied by k, the volumes are multiplied by k3.
- Write down the length scale factor. It is the number that every length is multiplied by.k = 3
- Work out the volume scale factor. Volume has three lengths in it (length × width × height), so it is multiplied by k three times: k3.volume scale factor = 33 = 3 × 3 × 3 = 27
- Multiply the original volume by the volume scale factor. The new value is the old value multiplied by the scale factor.5 × 27 = 135 cm3
Check. Divide back: 135 ÷ 27 = 5. ✓
Watch out. Do not multiply the area or volume by k. Area uses k2, volume uses k3.
Key words. The length scale factor k says how many times longer every length becomes. The area scale factor is k2 and the volume scale factor is k3. These work only for similar shapes or solids.
Method. If the lengths of two similar shapes are in the ratio a : b, then the areas are in the ratio a2 : b2 and the volumes are in the ratio a3 : b3.
- Write the length ratio in simplest form. Divide both numbers by their highest common factor.Lengths: 1 : 2
- Square both numbers for the area ratio. The area scale factor is the square of the length scale factor.Areas: 12 : 22 = 1 : 4
- Cube both numbers for the volume ratio. The volume scale factor is the cube of the length scale factor.Volumes: 13 : 23 = 1 : 8
Check. Test with a = 1, b = 2: a cube of side 1 has volume 1 and a cube of side 2 has volume 8. ✓
Watch out. Square (or cube) each number in the ratio, not just one of them.
Key words. The length scale factor k says how many times longer every length becomes. The area scale factor is k2 and the volume scale factor is k3. These work only for similar shapes or solids.
Method. If the lengths of two similar shapes are in the ratio a : b, then the areas are in the ratio a2 : b2 and the volumes are in the ratio a3 : b3.
- Write the length ratio in simplest form. Divide both numbers by their highest common factor.Lengths: 2 : 3
- Square both numbers for the area ratio. The area scale factor is the square of the length scale factor.Areas: 22 : 32 = 4 : 9
- Cube both numbers for the volume ratio. The volume scale factor is the cube of the length scale factor.Volumes: 23 : 33 = 8 : 27
Check. Test with a = 2, b = 3: a cube of side 2 has volume 8 and a cube of side 3 has volume 27. ✓
Watch out. Square (or cube) each number in the ratio, not just one of them.
Key words. The length scale factor k says how many times longer every length becomes. The area scale factor is k2 and the volume scale factor is k3. These work only for similar shapes or solids.
Method. If the lengths of two similar shapes are in the ratio a : b, then the areas are in the ratio a2 : b2 and the volumes are in the ratio a3 : b3.
- Write the length ratio in simplest form. Divide both numbers by their highest common factor.Lengths: 3 : 4
- Square both numbers for the area ratio. The area scale factor is the square of the length scale factor.Areas: 32 : 42 = 9 : 16
- Cube both numbers for the volume ratio. The volume scale factor is the cube of the length scale factor.Volumes: 33 : 43 = 27 : 64
Check. Test with a = 3, b = 4: a cube of side 3 has volume 27 and a cube of side 4 has volume 64. ✓
Watch out. Square (or cube) each number in the ratio, not just one of them.
Key words. The length scale factor k says how many times longer every length becomes. The area scale factor is k2 and the volume scale factor is k3. These work only for similar shapes or solids.
Method. If the lengths of two similar shapes are in the ratio a : b, then the areas are in the ratio a2 : b2 and the volumes are in the ratio a3 : b3.
- Write the length ratio in simplest form. Divide both numbers by their highest common factor.Lengths: 2 : 5
- Square both numbers for the area ratio. The area scale factor is the square of the length scale factor.Areas: 22 : 52 = 4 : 25
- Cube both numbers for the volume ratio. The volume scale factor is the cube of the length scale factor.Volumes: 23 : 53 = 8 : 125
Check. Test with a = 2, b = 5: a cube of side 2 has volume 8 and a cube of side 5 has volume 125. ✓
Watch out. Square (or cube) each number in the ratio, not just one of them.
Method. The area scale factor is the square of the length scale factor.
- Find the area scale factor. Area is multiplied by k2.32 = 9
- Multiply the area. 5 × 9 = 45 m2
Check. 45 ÷ 9 = 5. ✓
Watch out. Square the scale factor for area.
Method. The area scale factor is the square of the length scale factor. Then change the units.
- Area scale factor. The length scale factor is 100, so the area scale factor is 1002.100 × 100 = 10 000
- Real deck area in cm². 60 × 10 000 = 600 000 cm2
- Change to m². 1 m2 = 100 × 100 = 10 000 cm2.600 000 ÷ 10 000 = 60 m2
Check. Another way: 60 cm² × 10 000 = 60 m² since 1 m² = 10 000 cm², so the numbers match. ✓
Watch out. 1 m² is 10 000 cm², not 100 cm².
Method. Capacity is a volume, so use the volume scale factor k3.
- Find the length scale factor. 20 ÷ 10.k = 2
- Find the volume scale factor. Cube the length scale factor.23 = 8
- Multiply the capacity. 300 × 8 = 2400 ml
Check. 2400 ÷ 8 = 300. ✓
Watch out. Capacity is volume: cube the scale factor, do not just double.
Method. Work out the volume before and after doubling the lengths.
- Original volume. Volume = length × width × height.2 × 3 × 4 = 24 cm3
- Doubled lengths. Each length is multiplied by 2.4 cm by 6 cm by 8 cm
- New volume. 4 × 6 × 8 = 192 cm3
- Compare. 192 ÷ 24 = 8.The volume is multiplied by 8 = 23, not 2.
Check. Each of the three lengths doubles, so the volume doubles three times: 2 × 2 × 2 = 8. ✓
Watch out. A volume is made from three lengths, so it grows by k3.
Method. Surface area is an area, so it is multiplied by k2.
- Area scale factor. The length scale factor is 2.22 = 4
- New surface area. 50 × 4 = 200 cm2
- Compare with Sophie. Sophie multiplied by 2 only.100 cm2 is wrong; the answer is 200 cm2.
Check. 200 ÷ 4 = 50. ✓
Watch out. Only lengths are multiplied by k.
Method. The statues are made of the same stone, so their masses are in the same ratio as their volumes. Find the length scale factor, then cube it.
- Make the units the same. Change 1.2 m to centimetres. 1 m = 100 cm.1.2 × 100 = 120 cm
- Find the length scale factor. Large height ÷ small height.120 ÷ 40 = 3
- Find the volume (mass) scale factor. Cube the length scale factor, since mass is proportional to volume.33 = 27
- Find the mass of the large statue. Multiply the small statue’s mass by 27.2 × 27 = 54 kg
Check. Tripling every length makes 27 little statues fit in the big one: 27 × 2 kg = 54 kg. ✓
Watch out. The mass does not go up by 3 (the length factor) or 9 (the area factor), but by 27.
Exercise 21K Area and volume: problem solving
Non-calculatorMethod. The amount of pizza depends on the area. Find how many times bigger the area of the large pizza is, then compare with the price.
- Find the length scale factor. Diameter goes from 20 cm to 30 cm.30 ÷ 20 = 1.5
- Find the area scale factor. Area is multiplied by k2.1.52 = 1.5 × 1.5 = 2.25
- Find the price scale factor. £12 ÷ £6.12 ÷ 6 = 2
- Compare. The large pizza has 2.25 times as much pizza for 2 times the price.The 30 cm pizza is better value.
Check. Price per square unit: £6 for an area of 1 unit, £12 for 2.25 units is £12 ÷ 2.25 = £5.33 per unit. That is less than £6. ✓
Watch out. Do not compare the diameters (1.5 times) with the prices (2 times): the amount of food goes with the area, not the diameter.
Method. The paint needed is in proportion to the area. Find the area scale factor.
- Area scale factor. Lengths are doubled, so areas are multiplied by 22.22 = 4
- Multiply the paint. Four times the area needs four times the paint.3 × 4 = 12 litres
Check. Area of the second wall = 12 × 4 = 48 m², and 48 ÷ 12 = 4 lots of 3 litres. ✓
Watch out. Doubling the lengths of a wall does not just double the paint.
Key words. The length scale factor k says how many times longer every length becomes. The area scale factor is k2 and the volume scale factor is k3. These work only for similar shapes or solids.
Method. The model and the real object are similar, with length scale factor 50 (from model to real). Use 502 for areas and 503 for volumes.
- Area scale factor. Square the length scale factor.502 = 2500
- Real surface area. Multiply the model’s area by 2500.200 × 2500 = 500 000 cm2
- Change to m². 1 m2 = 10 000 cm2.500 000 ÷ 10 000 = 50 m2
- Volume scale factor. Cube the length scale factor.503 = 125 000
- Real volume. Multiply the model’s volume by 125 000.120 × 125 000 = 15 000 000 cm3
- Change to m³. 1 m3 = 1 000 000 cm3.15 000 000 ÷ 1 000 000 = 15 m3
Check. A length of 1 cm on the model is 50 cm = 0.5 m for real. 0.52 = 0.25 m2 per cm2, and 200 × 0.25 = 50. ✓
Watch out. 1 m2 = 10 000 cm2 and 1 m3 = 1 000 000 cm3.
Key words. The length scale factor k says how many times longer every length becomes. The area scale factor is k2 and the volume scale factor is k3. These work only for similar shapes or solids.
Method. The model and the real object are similar, with length scale factor 50 (from model to real). Use 502 for areas and 503 for volumes.
- Area scale factor. Square the length scale factor.502 = 2500
- Real surface area. Multiply the model’s area by 2500.200 × 2500 = 500 000 cm2
- Change to m². 1 m2 = 10 000 cm2.500 000 ÷ 10 000 = 50 m2
- Volume scale factor. Cube the length scale factor.503 = 125 000
- Real volume. Multiply the model’s volume by 125 000.120 × 125 000 = 15 000 000 cm3
- Change to m³. 1 m3 = 1 000 000 cm3.15 000 000 ÷ 1 000 000 = 15 m3
Check. A length of 1 cm on the model is 50 cm = 0.5 m for real. 0.52 = 0.25 m2 per cm2, and 200 × 0.25 = 50. ✓
Watch out. 1 m2 = 10 000 cm2 and 1 m3 = 1 000 000 cm3.
Method. Work out the surface area of each cube, then simplify the ratio. A cube has 6 square faces.
- Surface area of the small cube. Each face has area 2 × 2 = 4. There are 6 faces.6 × 4 = 24 cm2
- Surface area of the large cube. Each face has area 6 × 6 = 36.6 × 36 = 216 cm2
- Write the ratio and simplify. Divide both by 24.24 : 216 = 1 : 9
Check. The lengths are in the ratio 2 : 6 = 1 : 3, and 12 : 32 = 1 : 9. ✓
Watch out. Remember there are 6 faces on a cube.
Method. Work out the circumference and the area of both circles and compare.
- Original circle, radius 3 cm. Circumference = 2πr and area = πr2.Circumference = 2 × π × 3 = 6π cm
Area = π × 32 = 9π cm2 - Doubled circle, radius 6 cm. Use the same formulae.Circumference = 2 × π × 6 = 12π cm
Area = π × 62 = 36π cm2 - Compare. 12π ÷ 6π = 2 but 36π ÷ 9π = 4.The circumference doubles but the area is multiplied by 4, so Dan is wrong.
Check. The length scale factor is 2, so the area scale factor is 22 = 4. ✓
Watch out. Circumference is a length (k); area is k2.