[discia] GCSE Maths › Geometry & Measures › Sector Areas and Arc Lengths

Sector Areas and Arc Lengths

discia.org 10 questions · 32 marks
Name Date
  1. Either paper 1 mark

    Work out the arc length, in metres, of a semicircle of diameter 48 metres.

    Circle your answer.

    \(576\pi\qquad 24\pi\qquad 48\pi\qquad 288\pi\)

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    \(24\pi\) metres
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  2. Either paper 2 marks

    Here is a sector of a circle with centre \(O\).

    ABO6 cm60°

    The radius is \(6\) cm and the angle of the sector is \(60^\circ\).

    Work out the length of the arc \(AB\). Give your answer in terms of \(\pi\).

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    \(2\pi\ \text{cm}\)
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  3. Either paper 2 marks

    The diagram shows a sector of a circle with centre \(O\) and radius \(4\) cm.

    ABO4 cm135°

    The angle of the sector is \(135^\circ\).

    Find the area of the sector. Give your answer in terms of \(\pi\).

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    \(6\pi\ \text{cm}^2\)
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  4. Either paper 3 marks
    (a)

    The shaded sector \(AOB\) has an angle of \(135^\circ\).

    ABO8 cm135°

    Write down the size of the reflex angle \(AOB\).

    (b)

    Find the area of the unshaded part of the circle. Give your answer in terms of \(\pi\).

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    (a) \(225^\circ\)
    (b) \(40\pi\ \text{cm}^2\)
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  5. Either paper 3 marks

    The diagram shows a sector \(AOB\) of a circle with centre \(O\).

    ABO8 cm90°

    \(OA = OB = 8\) cm and angle \(AOB = 90^\circ\).

    Find the perimeter of the sector. Give your answer in terms of \(\pi\).

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    \((4\pi + 16)\ \text{cm}\)
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  6. Either paper 3 marks

    The diagram shows a sector of a circle with centre \(O\) and radius \(12\) cm.

    ABO12 cmx°

    The length of the arc \(AB\) is \(4\pi\) cm.

    Calculate the size of the angle marked \(x\).

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    \(60^\circ\)
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  7. Either paper 5 marks

    Shape A is a circle with radius \(3\) cm.

    Shape B is a sector of a circle with radius \(5\) cm.

    3 cm 225°5 cm

    The shapes are not drawn accurately.

    Which shape has the greater area, A or B?

    You must show your working.

    Show answer
    B: \(15.625\pi\ \text{cm}^2 > 9\pi\ \text{cm}^2\)
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  8. Calculator 4 marks

    Here is a circle, centre \(O\).

    xO

    The shaded part of the circle is the part of the circle outside angle x. The diagram is not accurately drawn.

    (a)

    The circle has a circumference of \(11\pi\) cm.

    Assume that angle \(x\) is \(120^\circ\).

    Find the shaded area. Give your answer as a decimal, correct to 2 decimal places.

    (b) In fact, angle \(x\) is larger than \(120^\circ\). What does this mean about the shaded area? Tick one box.
    • ☐ It is smaller than the answer to part (a)
    • ☐ It is the same as the answer to part (a)
    • ☐ It is larger than the answer to part (a)
    • ☐ It could be any of the above
    Show answer
    (a) \(63.36\ \text{cm²}\)
    (b) It is smaller than the answer to part (a)
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  9. Either paper 5 marks

    Two wire shapes make a hanging decoration.

    The shapes are a circle with radius \(20\) mm and a quarter circle. The quarter circle is the shaded part of circle \(P\) in the second diagram, and the rest of circle \(P\) is not wire. The wire for the quarter circle runs along its arc and along both of its straight edges.

    O20 mm P3π mm

    The diagrams are not drawn accurately.

    The arc of the quarter circle has length \(3\pi\) mm.

    (a)

    Show that the radius of the quarter circle is \(6\) mm.

    (b)

    Work out the total length of wire used for the hanging decoration.

    Write your answer as an exact length in the form \(a\pi + b\), with \(a\) and \(b\) whole numbers.

    Show answer
    (a) \(r = 6\ \text{mm}\)
    (b) \((43\pi + 12)\ \text{mm}\)
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  10. Either paper 4 marks

    \(OAB\) is a sector of a circle with centre \(O\) and radius \(6\) cm.

    ABO6 cmDiagram NOT accurately drawn

    The area of the sector is \(39.4\) cm\(^2\).

    Calculate the perimeter of the sector. Give your answer correct to 3 significant figures.

    Show answer
    \(25.1\ \text{cm}\)
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discia.org Sector Areas and Arc Lengths · sheet 1
Answers Sector Areas and Arc Lengths · sheet 1
  1. \(24\pi\) metres
  2. \(2\pi\ \text{cm}\)
  3. \(6\pi\ \text{cm}^2\)
  4. (a) \(225^\circ\)
    (b) \(40\pi\ \text{cm}^2\)
  5. \((4\pi + 16)\ \text{cm}\)
  6. \(60^\circ\)
  7. B: \(15.625\pi\ \text{cm}^2 > 9\pi\ \text{cm}^2\)
  8. (a) \(63.36\ \text{cm²}\)
    (b) It is smaller than the answer to part (a)
  9. (a) \(r = 6\ \text{mm}\)
    (b) \((43\pi + 12)\ \text{mm}\)
  10. \(25.1\ \text{cm}\)
discia.org Sector Areas and Arc Lengths · sheet 1