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Maths · Similarity and congruence

Similarity in 3D solids

Use length, area and volume scale factors for similar solids, including cones, cylinders and frustums.

  • Higher
  • 6 key terms
  • All boards
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Warm-up

Answer each one, then check.

  1. 1

    Work out \(2^3\).

    Show answerHide answer

    8

  2. 2

    Work out \(1.5^3\).

    Show answerHide answer

    3.375

  3. 3

    What is the area scale factor if \(k = 3\)?

    Show answerHide answer

    9

  4. 4

    Work out \(\sqrt[3]{27}\).

    Show answerHide answer

    3

  5. 5

    What is the formula for the volume of a cone?

    Show answerHide answer

    \(\dfrac{1}{3}\pi r^2 h\)

Learning Objectives

  1. 1Use length, area and volume scale factors.
  2. 2Find a volume, capacity or mass of a similar solid.
  3. 3Work out the length ratio from a volume ratio.
  4. 4Solve problems with similar cones, cylinders and frustums.

Scale Factors for Similar Solids

If the length scale factor is \(k\).

  • Lengths

    Scale factor: \(k\). If \(k = 2\): 2

  • Areas (including surface area)

    Scale factor: \(k^2\). If \(k = 2\): 4

  • Volumes (and capacity, mass)

    Scale factor: \(k^3\). If \(k = 2\): 8

Volume of a Similar Cone

Cone A has height 6 cm and volume 40 cm³. Cone B is similar to A and has height 9 cm. Find the volume of cone B.

Show the solutionHide the solution
  1. 1 Length scale factor \(\dfrac{9}{6} = 1.5\)
  2. 2 Volume scale factor \(1.5^3 = 3.375\)
  3. 3 Volume of B \(40 \times 3.375\)

Answer135 cm³

A Model and the Real Thing

A model statue is 20 cm tall and has mass 2 kg. The real statue is 1.2 m tall and made of the same material. Find the mass of the real statue.

Show the solutionHide the solution
  1. 1 Length scale factor \(120 \div 20 = 6\)
  2. 2 Volume (and mass) scale factor \(6^3 = 216\)
  3. 3 Mass \(2 \times 216\)

Answer432 kg

A Frustum

A cone of height 15 cm has volume 500 cm³. A small cone of height 6 cm is cut off the top, leaving a frustum. Find the volume of the frustum.

Show the solutionHide the solution
  1. 1 The small cone is similar to the large cone Length scale factor \(\dfrac{6}{15} = 0.4\)
  2. 2 Volume scale factor \(0.4^3 = 0.064\)
  3. 3 Volume of the small cone \(500 \times 0.064 = 32\)
  4. 4 Frustum \(=\) large cone minus small cone \(500 - 32\)

Answer468 cm³

From Volumes to Lengths

Two similar solids have volumes 8 cm³ and 125 cm³. Find the ratio of their lengths and of their surface areas.

Show the solutionHide the solution
  1. 1 Take the cube root of the volumes \(\sqrt[3]{8} : \sqrt[3]{125} = 2 : 5\)
  2. 2 Lengths \(2 : 5\)
  3. 3 Surface areas: square the ratio \(4 : 25\)

AnswerLengths \(2 : 5\); surface areas \(4 : 25\).

Which Scale Factor?

Use \(k\) (lengths)

  • Heights, radii, edges, perimeters.
  • Anything measured in a straight line.
  • Read directly from the two shapes.

Use \(k^2\) or \(k^3\)

  • \(k^2\): area, surface area, cross-section.
  • \(k^3\): volume, capacity, mass (same material).
  • Cube or square root to go backwards.

Bottle Sizes

A small bottle is 10 cm tall and holds 250 ml. A similar large bottle is 15 cm tall. Find the capacity of the large bottle. A label on the small bottle has area 40 cm²; find the label area on the large bottle.

1. Find k.

2. Cube for volume.

3. Square for area.

A good answer shows: Scale factor 1.5. Capacity \(250 \times 1.5^3 = 843.75\) ml, about 844 ml. Label \(40 \times 1.5^2 = 90\) cm².

Can I...?

  1. 1State the scale factors \(k\), \(k^2\), \(k^3\).
  2. 2Find a volume of a similar solid.
  3. 3Find a mass using a volume factor.
  4. 4Find the surface area of a similar solid.
  5. 5Use a cube root to find a length ratio.
  6. 6Find the volume of a frustum.
  7. 7Choose the right scale factor.
  8. 8Show clear working.

Summary & Exam Focus

  • Lengths \(k\), areas \(k^2\), volumes \(k^3\).
  • For the same material, mass scales like volume.
  • From a volume ratio take the cube root for lengths.
  • A frustum is a large cone minus a similar small cone.

Exam focus

Cone A has height 6 cm and volume 40 cm³. Cone B is similar to A and has height 9 cm. Work out the volume of cone B. (3 marks) (3 marks)

Find the length scale factor first. Cube it for volume. Do not square it.

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Similar solids
Solids with the same shape and all lengths in the same ratio.
Volume scale factor
\(k^3\).
Surface area scale factor
\(k^2\).
Frustum
A cone or pyramid with the top cut off parallel to the base.
Capacity
The volume a container holds.
Density
Mass per unit volume; the same in similar solids of the same material.

Practice questions

Have a go at each one before you open its answer.

  1. Question 1 Work out 3 marks

    Cone A and cone B are mathematically similar. The height of cone A is 6 cm and its volume is 40 cm³. The height of cone B is 9 cm. Work out the volume of cone B.

    Two similar cones: A with height 6 cm and volume 40 cm cubed, and a larger cone B with height 9 cm.
    Show answerHide answer

    Model answer

    The length scale factor is \(\dfrac{9}{6} = 1.5\). The volume scale factor is \(1.5^3 = 3.375\). The volume is \(40 \times 3.375 = 135\) cm³.

    Mark scheme

    • Length scale factor 1.5 — M1
    • \(1.5^3 = 3.375\) — M1
    • 135 — A1
  2. Question 2 Work out 3 marks

    A model of a statue is 20 cm tall and has a mass of 2 kg. The real statue is 1.2 m tall and is made from the same material. Work out the mass of the real statue.

    Show answerHide answer

    Model answer

    The length scale factor is \(120 \div 20 = 6\). The volume scale factor is \(6^3 = 216\). The mass is \(2 \times 216 = 432\) kg.

    Mark scheme

    • Scale factor 6 — M1
    • \(6^3 = 216\) — M1
    • 432 — A1
  3. Question 3 Work out 4 marks

    A cone has a height of 15 cm and a volume of 500 cm³. A smaller cone of height 6 cm is cut off the top, leaving a frustum. The small cone is similar to the large cone. Work out the volume of the frustum.

    Show answerHide answer

    Model answer

    The length scale factor is \(\dfrac{6}{15} = 0.4\) and the volume scale factor is \(0.4^3 = 0.064\). The small cone has volume \(500 \times 0.064 = 32\) cm³. The frustum is \(500 - 32 = 468\) cm³.

    Mark scheme

    • Scale factor 0.4 — M1
    • \(0.4^3 = 0.064\) — M1
    • Small cone 32 — A1
    • 468 — A1
  4. Question 4 Work out 3 marks

    Two similar solids have surface areas in the ratio \(4 : 9\). Work out the ratio of their volumes.

    Show answerHide answer

    Model answer

    The length ratio is \(\sqrt{4} : \sqrt{9} = 2 : 3\). The volume ratio is \(2^3 : 3^3 = 8 : 27\).

    Mark scheme

    • Length ratio \(2 : 3\) — M1
    • Cubing — M1
    • \(8 : 27\) — A1
  5. Question 5 Work out 3 marks

    A small bottle is 10 cm tall and holds 250 ml of juice. A similar large bottle is 15 cm tall. Work out the capacity of the large bottle. Give your answer to the nearest ml.

    Show answerHide answer

    Model answer

    The scale factor is 1.5, so the volume scale factor is \(1.5^3 = 3.375\). The capacity is \(250 \times 3.375 = 843.75 \approx 844\) ml.

    Mark scheme

    • Scale factor 1.5 — M1
    • \(250 \times 1.5^3\) — M1
    • 844 — A1
  6. Question 6 Work out 3 marks

    Two similar solids have volumes of 8 cm³ and 125 cm³. The surface area of the smaller solid is 20 cm². Work out the surface area of the larger solid.

    Show answerHide answer

    Model answer

    The length ratio is \(\sqrt[3]{8} : \sqrt[3]{125} = 2 : 5\), so the area scale factor is \(\left(\dfrac{5}{2}\right)^2 = 6.25\). The surface area is \(20 \times 6.25 = 125\) cm².

    Mark scheme

    • Length scale factor \(\dfrac{5}{2}\) — M1
    • Area scale factor 6.25 — M1
    • 125 — A1

Quick check

  1. The length scale factor of two similar solids is 2. What is the volume scale factor?

    1. A2
    2. B4
    3. C8
    4. D6
    Show answerHide answer

    C: 8

    \(2^3 = 8\).

  2. A model is 1/10 the size of a real object. The model's volume is what fraction of the real volume?

    1. A\(\dfrac{1}{10}\)
    2. B\(\dfrac{1}{1000}\)
    3. C\(\dfrac{1}{100}\)
    4. D\(\dfrac{1}{30}\)
    Show answerHide answer

    B: \(\dfrac{1}{1000}\)

    \(\left(\dfrac{1}{10}\right)^3 = \dfrac{1}{1000}\).

  3. Two similar solids have volume ratio \(1 : 27\). What is the length ratio?

    1. A\(1 : 3\)
    2. B\(1 : 9\)
    3. C\(1 : 27\)
    4. D\(1 : 5.2\)
    Show answerHide answer

    A: \(1 : 3\)

    \(\sqrt[3]{27} = 3\), so \(1 : 3\).

  4. Two similar shapes of the same material have length ratio \(1 : 2\). What is the ratio of their masses?

    1. A\(1 : 2\)
    2. B\(1 : 4\)
    3. C\(1 : 6\)
    4. D\(1 : 8\)
    Show answerHide answer

    D: \(1 : 8\)

    Mass scales like volume: \(1 : 8\).

  5. To find the volume of a frustum you...

    1. AAdd the two cones
    2. BSubtract the small cone from the large cone
    3. CMultiply the cones
    4. DHalve the large cone
    Show answerHide answer

    B: Subtract the small cone from the large cone

    Subtract the small similar cone from the large cone.

  6. The surface areas of two similar solids are 25 cm² and 100 cm². If the smaller has volume 30 cm³, what is the volume of the larger?

    1. A120 cm³
    2. B60 cm³
    3. C240 cm³
    4. D480 cm³
    Show answerHide answer

    C: 240 cm³

    The length ratio is \(1 : 2\), so the volume is \(30 \times 8 = 240\) cm³.

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