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GCSE Maths · EdexcelChapter 21 · Transformations, congruence and similarity
GCSE Maths · Transformations and Similarity · Higher · Lesson 21.9

GCSE Maths: Area and volume scale factors (Higher)

Part of the GCSE Maths transformations and similarity chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)

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21.9

Area and volume scale factors

Edexcel 1MA1: G19, R12
Grade 4-9

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.

side 1area 1side 2area 4side 3area 9
Doubling the side of a square multiplies its area by 22 = 4. Trebling multiplies it by 32 = 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.

111222333
Cubes with side 1, 2 and 3 have volumes 1, 8 and 27: the volume is multiplied by k3.

This works for any similar shapes, not only squares and cubes. If the length scale factor is k:

QuantityScale 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.

Working backwards. If you are given the area (or volume) scale factor, take the square root (or cube root) to find the length scale factor. For example, if the areas are in the ratio 9 : 25, the lengths are in the ratio 3 : 5, and the volumes are in the ratio 27 : 125.

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.
Worked example 21.9.1
Two similar triangles have a length scale factor of 3. The area of the smaller triangle is 12 cm2. Work out the area of the larger triangle.

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.

  1. Write down the length scale factor. It is the number that every length is multiplied by.
    k = 3
  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 = 32 = 3 × 3 = 9
  3. 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.

Worked example 21.9.2
Two similar solids have a length scale factor of 2. The volume of the smaller solid is 15 cm3. Work out the volume of the larger solid.

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.

  1. Write down the length scale factor. It is the number that every length is multiplied by.
    k = 2
  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
  3. 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.

Worked example 21.9.3
The lengths of two similar shapes are in the ratio 2 : 5. Write down the ratio of (a) their areas, (b) their volumes.

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.

  1. Write the length ratio in simplest form. Divide both numbers by their highest common factor.
    Lengths: 2 : 5
  2. Square both numbers for the area ratio. The area scale factor is the square of the length scale factor.
    Areas: 22 : 52 = 4 : 25
  3. 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.

Worked example 21.9.4
Two similar shapes have areas 20 cm2 and 80 cm2. A side of the smaller shape is 7 cm. Work out the length of the matching side of the larger shape.

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 areas are in the ratio k2. Find the ratio of the areas, then take the square root to get the length scale factor k.

  1. Find the area scale factor. Area scale factor = new area ÷ old area.
    80 ÷ 20 = 4
  2. Take the square root to find the length scale factor. The area scale factor is k2, so k is its square root.
    k = √4 = 2
  3. Use k to find the new length. Lengths are multiplied by the length scale factor k.
    7 × 2 = 14 cm

Check. Put it back: k2 = 22 = 4, and 20 × 4 = 80. ✓

Watch out. The square root goes from area to length. Do not use the area ratio as the length scale factor.

Worked example 21.9.5
The areas of two similar shapes are in the ratio 9 : 25. Work out the ratio of their volumes.

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. Go from the area ratio to the length ratio, then to the volume ratio.

  1. Find the length ratio. The area ratio is the length ratio raised to the power 2, so take the square root of each number.
    Areas 9 : 25
    Lengths: √9 : √25 = 3 : 5
  2. Raise the length ratio to the power 3. The volume ratio is the length ratio raised to the power 3.
    33 : 53 = 27 : 125
  3. Write the answer. Name what the ratio is for.
    Volumes: 27 : 125

Check. Go back: 27 : 125 with the cube root gives 3 : 5, the length ratio. ✓

Watch out. You cannot go straight from area to volume by multiplying: always go through the length ratio.

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-calculator
1
FEach pair of shapes is similar. The length scale factor from the small shape to the large shape is given. Work out the area or volume of the large shape.
(a)Length scale factor 2. Area of small shape = 5 cm². Area of large shape?
(b)Length scale factor 3. Area of small shape = 7 cm². Area of large shape?
(c)Length scale factor 2. Volume of small solid = 9 cm³. Volume of large solid?
(d)Length scale factor 3. Volume of small solid = 4 cm³. Volume of large solid?
2
FA square has sides of length 3 cm. A larger square has sides of length 6 cm. (a) Work out the area of each square. (b) Write the ratio of the areas in its simplest form. (c) Write down the ratio of the lengths, and compare it with the ratio of the areas.
(a)Areas of the squares
(b)Ratio of the areas
(c)Ratio of the lengths, and comparison
3
FA cube has a volume of 5 cm³. A larger cube has lengths that are 3 times as long. Work out the volume of the larger cube.
4
FThe lengths of two similar shapes are in the given ratio. Write down the ratio of their areas and the ratio of their volumes.
(a)1 : 2
(b)2 : 3
(c)3 : 4
(d)4 : 10
5
PSA lawn has an area of 5 m². A similar lawn has all its lengths 3 times as long. Work out the area of the larger lawn.
6
PSA model of a ship is 20 cm long. The real ship is 100 times as long. The model has a deck area of 60 cm². Work out the deck area of the real ship in m².
7
PSTwo similar bottles have heights 10 cm and 20 cm. The smaller bottle holds 300 ml of juice. How much does the larger bottle hold?
8
RTom says: “If I double all the lengths of a cuboid, its volume doubles.” Explain why Tom is wrong. Use a cuboid 2 cm by 3 cm by 4 cm.
9
RA cylinder has a surface area of 50 cm². It is enlarged with a scale factor of 2. Sophie says the new surface area is 100 cm². Is she right? Explain.
10
FTwo similar shapes have areas 20 cm² and 80 cm². A side of the smaller shape is 7 cm. Work out the length of the matching side of the larger shape.
11
FTwo similar solids have volumes 8 cm³ and 125 cm³. A length on the smaller solid is 6 cm. Work out the matching length on the larger solid.
12
FThe areas (or volumes) of two similar shapes are given as a ratio. Find the ratio asked for.
(a)Areas 4 : 9. Write the ratio of the volumes.
(b)Areas 9 : 25. Write the ratio of the volumes.
(c)Volumes 8 : 27. Write the ratio of the areas.
(d)Volumes 27 : 64. Write the ratio of the lengths.
13
PSTwo similar tins of beans have heights 8 cm and 12 cm. The smaller tin holds 400 ml and costs 80p. The larger tin costs £2.50. Work out the volume of the larger tin. Which tin is better value for money? Show your working.
14
RChallenge. Two similar statues are made of the same stone. The small statue is 40 cm tall and weighs 2 kg. The large statue is 1.2 m tall. Work out the weight of the large statue.

Exercise 21K  Area and volume: problem solving

Non-calculator
1
PSA pizza with a diameter of 20 cm costs £6. A similar pizza has a diameter of 30 cm and costs £12. Which pizza is better value? Explain. (Both pizzas have the same thickness and the same toppings.)
2
PSA wall of area 12 m² needs 3 litres of paint. A similar wall has every length twice as long. How much paint does it need?
3
PSA model of a building is made to a scale of 1 : 50. The model has a surface area of 200 cm² and a volume of 120 cm³. Work out (a) the surface area, in m², of the real building, (b) the volume, in m³, of the real building.
(a)(a) Surface area of the real building
(b)(b) Volume of the real building
4
FTwo cubes have sides of 2 cm and 6 cm. Work out the ratio of their surface areas, in its simplest form.
5
RDan says: “If I double the radius of a circle, the circumference doubles, so the area must double too.” Show that Dan is wrong. Use a circle of radius 3 cm and leave your answers in terms of π.
6
PSTwo similar cones have heights of 6 cm and 9 cm. The smaller cone has a curved surface area of 80 cm² and a volume of 150 cm³. Work out the curved surface area and the volume of the larger cone.
(a)Curved surface area of the larger cone
(b)Volume of the larger cone
7
PSTwo similar jars are made of the same glass. The small jar is 10 cm tall and holds 400 ml. The large jar holds 1350 ml. Work out the height of the large jar.
8
PSTwo similar triangles have areas of 18 cm² and 50 cm². The perimeter of the smaller triangle is 24 cm. Work out the perimeter of the larger triangle.
9
RA cube has side a. A larger cube has side ka. Show that the ratio of their volumes is 1 : k3.
10
RChallenge. A solid is enlarged so that its surface area increases by 125%. Work out the percentage increase in its volume.

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