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Similar Shapes (Area and Volume) worksheet

10 GCSE practice questions with answers, free to print. Every sheet is generated, so you can make a fresh one whenever you need it.

10 questions 34 marks Sheet 1
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  1. 3 marks

    Shape \(A\) and shape \(B\) are mathematically similar triangles.

    The base of shape \(A\) is \(3\) cm and the base of shape \(B\) is \(6\) cm.

    The area of shape \(A\) is \(21\) cm².

    (a)

    Write down the scale factor of the enlargement from shape \(A\) to shape \(B\).

    (b)

    Calculate the area of shape \(B\).

  2. 3 marks

    Here is solid \(A\), a cone of height \(6\) cm.

    Solid A, a cone drawn in isometric projection and not to scale, with its height labelled 6 cm6 cmADiagram NOT accurately drawn

    Solid \(B\) is a cone that is mathematically similar to solid \(A\). The height of solid \(B\) is \(18\) cm.

    The volume of solid \(A\) is \(66\) cm³.

    Work out the volume of solid \(B\).

  3. 4 marks

    Cuboid \(A\) and cuboid \(B\) are mathematically similar.

    Solid A, a cuboid drawn in isometric projection and not to scale, with its height labelled 4 cm4 cmADiagram NOT accurately drawn Solid B, a cuboid drawn in isometric projection and not to scale, with its height labelled 12 cm12 cmBDiagram NOT accurately drawn

    The surface area of cuboid \(A\) is \(52\) cm² and the volume of cuboid \(A\) is \(24\) cm³.

    (a)

    Work out the surface area of cuboid \(B\).

    (b)

    Work out the volume of cuboid \(B\).

  4. 4 marks

    Shape \(P\) and shape \(Q\) are mathematically similar.

    Every length on shape \(Q\) is \(3\) times the corresponding length on shape \(P\).

    The area of shape \(P\) is \(8\) cm².

    Marcus says:

    "The area of shape \(Q\) is \(24\) cm², because every length is \(3\) times as big."

    (a)

    Explain why Marcus is wrong.

    (b)

    Work out the correct area of shape \(Q\).

  5. 3 marks

    Shape \(A\) and shape \(B\) are mathematically similar hexagons.

    The area of shape \(A\) is \(16\) cm² and the area of shape \(B\) is \(36\) cm².

    The shortest side of shape \(A\) is \(4\) cm.

    Work out the shortest side of shape \(B\).

  6. 3 marks

    Two pyramids, \(A\) and \(B\), are mathematically similar.

    The volume of \(A\) is \(4\) cm³ and the volume of \(B\) is \(256\) cm³.

    The height of \(A\) is \(4\) cm.

    Work out the height of \(B\).

  7. 4 marks

    Idris makes a scale model of a monument.

    The model is \(50\) cm tall and the monument is \(4\) m tall. The model and the monument are mathematically similar, and they are made of the same material.

    The model has a mass of \(5\) kg.

    Work out the mass of the monument.

  8. 3 marks

    Shape \(A\) and shape \(B\) are mathematically similar pentagons.

    The shortest side of shape \(A\) is \(9\) cm and the shortest side of shape \(B\) is \(18\) cm.

    The area of shape \(A\) is \(16\) cm².

    (a)

    State the scale factor of the enlargement from shape \(A\) to shape \(B\).

    (b)

    Find the area of shape \(B\).

  9. 3 marks

    Here is solid \(A\), a cone of height \(10\) cm.

    Solid A, a cone drawn in isometric projection and not to scale, with its height labelled 10 cm10 cmADiagram NOT accurately drawn

    Solid \(B\) is a cone that is mathematically similar to solid \(A\). The height of solid \(B\) is \(20\) cm.

    The volume of solid \(A\) is \(110\) cm³.

    Find the volume of solid \(B\).

  10. 4 marks

    Cuboid \(A\) and cuboid \(B\) are mathematically similar.

    Solid A, a cuboid drawn in isometric projection and not to scale, with its height labelled 6 cm6 cmADiagram NOT accurately drawn Solid B, a cuboid drawn in isometric projection and not to scale, with its height labelled 12 cm12 cmBDiagram NOT accurately drawn

    The surface area of cuboid \(A\) is \(148\) cm² and the volume of cuboid \(A\) is \(120\) cm³.

    (a)

    Calculate the surface area of cuboid \(B\).

    (b)

    Calculate the volume of cuboid \(B\).

Answers

  1. (a) \(2\)
    (b) \(84\ \text{cm}^2\)
  2. \(1782\ \text{cm}^3\)
  3. (a) \(468\ \text{cm}^2\)
    (b) \(648\ \text{cm}^3\)
  4. (a) An area has two dimensions and both are multiplied by \(3\), so the area is multiplied by \(3^2 = 9\), not by \(3\).
    (b) \(72\ \text{cm}^2\)
  5. \(6\ \text{cm}\)
  6. \(16\ \text{cm}\)
  7. \(2560\ \text{kg}\)
  8. (a) \(2\)
    (b) \(64\ \text{cm}^2\)
  9. \(880\ \text{cm}^3\)
  10. (a) \(592\ \text{cm}^2\)
    (b) \(960\ \text{cm}^3\)

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