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Circles, arcs and sectors Cambridge IGCSE Mathematics revision

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In plain words

Wrap a tape round any circle, divide by the distance across, and you always get the same number: a little over 3. That number is π. Everything about circles comes from it.

Three things to know

  1. Circumference = π × diameter = 2πr. Area = πr².
  2. An arc is part of the circumference, and a sector is part of the area. For an angle θ at the centre, each is θ/360 of the whole: arc = θ/360 × 2πr, sector area = θ/360 × πr².
  3. If you're given the diameter, halve it before using πr².

Worked example

Find the area of a sector of radius 6 cm with an angle of 60°.

  1. The sector is 60/360 of the whole circle, which is one sixth.
  2. Area = 60/360 × π × 6² = 1/6 × 36π = 6π.
  3. = 18.8 cm² (to 3 significant figures).

Tips and tricks

  • Area has r squared. Circumference doesn't. "Area: r squared" is how to remember which is which.
  • Use the π button on your calculator, and round only at the end.
5 questions, about 2 minutes.

It lands in your notebook with its questions as flashcards.

Circles, arcs and sectors: 5 questions and answers

These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.

  1. What is the formula for the area of a circle?
    • 2πr
    • πd
    • πr² (the answer)
    • 2πr²

    Radius squared, times π.

  2. A circle has a radius of 3 cm. What is its circumference?
    • 3π cm
    • 6π cm (the answer)
    • 9π cm
    • 12π cm

    2 × π × 3.

  3. A circle has a diameter of 10 cm. What is its area?
    • 10π cm²
    • 25π cm² (the answer)
    • 100π cm²
    • 5π cm²

    The radius is 5, and 5² = 25.

  4. What fraction of a circle is a sector with an angle of 90°?
    • 1/2
    • 1/3
    • 1/4 (the answer)
    • 1/9

    90/360.

  5. A sector has radius 4 cm and angle 180°. What is its area?
    • 4π cm²
    • 8π cm² (the answer)
    • 16π cm²
    • 2π cm²

    Half of π × 4².

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