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Similar Shapes (Area and Volume)

discia.org 10 questions · 32 marks
Name Date
  1. Either paper 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 \(B\).

    Show answer
    (a) \(2\)
    (b) \(84\ \text{cm}^2\)
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  2. Either paper 1 mark
    \(A\) and \(B\) are similar solids.
    SolidLength (cm)
    \(A\)\(9\)
    \(B\)\(36\)
    Leila says, "The volume of \(B\) is \(64\) times the volume of \(A\) because the length of \(B\) is \(4\) times the length of \(A\)." Is she correct? Tick a box. ☐ Yes    ☐ No Give a reason for your answer.
    Show answer
    Yes. The lengths are in the ratio \(1 : 4\), so the volume scale factor is \(4^3 = 64\), and the volume of \(B\) is \(64\) times the volume of \(A\), which is what is claimed.
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  3. Either paper 3 marks

    Here is solid \(A\), a cone of height \(7\) cm.

    7 cmADiagram NOT accurately drawn

    Solid \(B\) is a cone that is mathematically similar to solid \(A\). The height of solid \(B\) is \(14\) cm.

    The volume of solid \(A\) is \(28\) cm³.

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

    Show answer
    \(224\ \text{cm}^3\)
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  4. Either paper 4 marks

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

    6 cmADiagram NOT accurately drawn 18 cmBDiagram NOT accurately drawn

    The surface area of cuboid \(A\) is \(88\) cm² and the volume of cuboid \(A\) is \(48\) cm³.

    (a)

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

    (b)

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

    Show answer
    (a) \(792\ \text{cm}^2\)
    (b) \(1296\ \text{cm}^3\)
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  5. Either paper 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 \(Q\).

    Show answer
    (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) \(72\ \text{cm}^2\)
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  6. Either paper 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 \(B\).

    Show answer
    \(6\ \text{cm}\)
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  7. Either paper 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 \(B\).

    Show answer
    \(16\ \text{cm}\)
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  8. Either paper 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.

    Show answer
    \(2560\ \text{kg}\)
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  9. Either paper 4 marks

    The diagram shows two similar solid triangular prisms, prism \(A\) and prism \(B\).

    h cmADiagram NOT accurately drawn 12.5 cmBDiagram NOT accurately drawn

    The volume of prism \(A\) is \(88\) cm³.

    The volume of prism \(B\) is \(1375\) cm³.

    For prism \(B\), the length of the base of the triangle is \(12.5\) cm and the area of the triangle is \(137.5\) cm².

    The height of the triangle for prism \(A\) is \(h\) cm.

    Work out the value of \(h\).

    Show answer
    \(8.8\ \text{cm}\)
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  10. Either paper 3 marks

    Triangles \(OPX\), \(OQY\) and \(ORZ\) are similar.

    \(OPQR\) and \(OXYZ\) are straight lines.

    OPQRZYX

    \(OP : PQ : QR = 1 : 1 : 3\)

    Show that

    area of triangle \(OPX\) : area of \(PQYX\) : area of \(QRZY\) \(= 1 : 3 : 21\)

    Show answer
    Triangle \(OPX\) : strip \(PQYX\) : strip \(QRZY\) \(= 1 : 4 - 1 : 25 - 4 = 1 : 3 : 21\)
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discia.org Similar Shapes (Area and Volume) · sheet 1
Answers Similar Shapes (Area and Volume) · sheet 1
  1. (a) \(2\)
    (b) \(84\ \text{cm}^2\)
  2. Yes. The lengths are in the ratio \(1 : 4\), so the volume scale factor is \(4^3 = 64\), and the volume of \(B\) is \(64\) times the volume of \(A\), which is what is claimed.
  3. \(224\ \text{cm}^3\)
  4. (a) \(792\ \text{cm}^2\)
    (b) \(1296\ \text{cm}^3\)
  5. (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) \(72\ \text{cm}^2\)
  6. \(6\ \text{cm}\)
  7. \(16\ \text{cm}\)
  8. \(2560\ \text{kg}\)
  9. \(8.8\ \text{cm}\)
  10. Triangle \(OPX\) : strip \(PQYX\) : strip \(QRZY\) \(= 1 : 4 - 1 : 25 - 4 = 1 : 3 : 21\)
discia.org Similar Shapes (Area and Volume) · sheet 1