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Maths · Area and volume

Sectors of circles

Work out the arc length, area and perimeter of a sector as a fraction of a whole circle, and at Higher tier the area of a segment.

  • 6 key terms
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Warm-up

Answer each one, then check.

  1. 1

    What is the circumference of a circle with radius 5 cm, in terms of \(\pi\)?

    Show answerHide answer

    \(10\pi\)

  2. 2

    What is the area of a circle with radius 6 cm, in terms of \(\pi\)?

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    \(36\pi\)

  3. 3

    Simplify \(\dfrac{90}{360}\).

    Show answerHide answer

    \(\dfrac{1}{4}\)

  4. 4

    How many degrees in a full turn?

    Show answerHide answer

    360

  5. 5

    What is the area of a triangle with base 8 and height 8?

    Show answerHide answer

    32

Learning Objectives

  1. 1Identify the arc and sector of a circle.
  2. 2Find arc length and sector area as a fraction of the whole circle.
  3. 3Find the perimeter of a sector.
  4. 4Find the area of a segment (Higher).

Sector Formulae

  • Fraction of the circle

    \(\dfrac{\theta}{360}\), where \(\theta\) is the angle at the centre in degrees.

  • Arc length

    \(\dfrac{\theta}{360} \times 2\pi r\): a fraction of the circumference.

  • Sector area

    \(\dfrac{\theta}{360} \times \pi r^2\): a fraction of the circle's area.

  • Sector perimeter

    Arc length \(+\ 2r\) (the two straight radii).

Arc Length and Sector Area

A sector has radius 6 cm and angle \(60^\circ\). Find the arc length and the area, in terms of \(\pi\).

Show the solutionHide the solution
  1. 1 Fraction of the circle \(\dfrac{60}{360} = \dfrac{1}{6}\)
  2. 2 Arc length \(\dfrac{1}{6} \times 2\pi \times 6 = 2\pi\)
  3. 3 Area \(\dfrac{1}{6} \times \pi \times 6^2 = 6\pi\)

AnswerArc length \(2\pi\) cm (6.28 cm) and area \(6\pi\) cm² (18.85 cm²).

Perimeter of a Sector

A sector has radius 5 cm and angle \(90^\circ\). Find its perimeter to 1 decimal place.

Show the solutionHide the solution
  1. 1 Arc length \(\dfrac{90}{360} \times 2\pi \times 5 = 2.5\pi = 7.854...\)
  2. 2 Add the two radii \(2 \times 5 = 10\)
  3. 3 Total \(7.854 + 10\)

Answer17.9 cm

Finding the Angle

A sector has radius 12 cm and an arc length of \(8\pi\) cm. Find the angle at the centre.

Show the solutionHide the solution
  1. 1 Arc length formula \(\dfrac{\theta}{360} \times 2\pi \times 12 = 8\pi\)
  2. 2 Simplify \(\dfrac{\theta}{360} \times 24\pi = 8\pi\)
  3. 3 Divide both sides by \(24\pi\) \(\dfrac{\theta}{360} = \dfrac{1}{3}\)
  4. 4 Solve \(\theta = 120\)

Answer\(120^\circ\)

Segment Area

A sector has radius 8 cm and angle \(90^\circ\). Find the area of the segment cut off by the chord, to 1 decimal place.

Show the solutionHide the solution
  1. 1 Sector area \(\dfrac{90}{360} \times \pi \times 8^2 = 16\pi = 50.27\)
  2. 2 Triangle area \(\dfrac{1}{2}r^2 \sin\theta\) \(\dfrac{1}{2} \times 8 \times 8 \times \sin 90^\circ = 32\)
  3. 3 Subtract \(50.27 - 32\)

Answer18.3 cm²

Sector or Segment?

The words are easy to mix up.

  • Sector

    The "pizza slice" between two radii and an arc.

  • Segment

    The region between a chord and an arc. It is a sector with a triangle cut off.

  • Segment area

    Sector area \(-\) triangle area, where the triangle has area \(\dfrac{1}{2}r^2\sin\theta\).

Slice the Clock

The minute hand of a clock is 15 cm long. Find the distance its tip travels in 20 minutes, and the area swept out by the hand in 20 minutes. Give both answers in terms of \(\pi\).

1. Find the fraction of a full turn.

2. Apply it to circumference and area.

A good answer shows: In 20 minutes the hand turns through \(120^\circ\), which is \(\dfrac{1}{3}\) of a circle. Distance \(= \dfrac{1}{3} \times 30\pi = 10\pi\) cm. Area \(= \dfrac{1}{3} \times 225\pi = 75\pi\) cm².

Can I...?

  1. 1Write the fraction \(\dfrac{\theta}{360}\).
  2. 2Find an arc length.
  3. 3Find a sector area.
  4. 4Find the perimeter of a sector.
  5. 5Find the angle from an arc length.
  6. 6Give answers in terms of \(\pi\).
  7. 7Find a segment area (Higher).
  8. 8Tell a sector from a segment.

Summary & Exam Focus

  • Arc length \(= \dfrac{\theta}{360} \times 2\pi r\).
  • Sector area \(= \dfrac{\theta}{360} \times \pi r^2\).
  • Sector perimeter \(=\) arc \(+\ 2r\).
  • Segment area \(=\) sector \(-\) triangle (Higher).

Exam focus

A sector of a circle has radius 9 cm and angle \(40^\circ\). Work out the length of the arc of the sector. Give your answer correct to 3 significant figures. (3 marks) (3 marks)

Write the fraction \(\dfrac{\theta}{360}\) first. Then multiply it by the circumference (arc) or the area (sector).

Key terms

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

Sector
A region of a circle between two radii and the arc joining their ends.
Arc
Part of the circumference of a circle.
Segment
The part of a circle cut off by a chord.
Chord
A straight line joining two points on a circle.
Angle at the centre
The angle between the two radii of a sector.
Fraction of a circle
\(\dfrac{\theta}{360}\).

Practice questions

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

  1. Question 1 Calculator 3 marks

    The diagram shows a sector of a circle of radius 9 cm. The angle at the centre is \(40^\circ\). Work out the length of the arc. Give your answer correct to 3 significant figures.

    A sector with radius 9 cm and angle 40 degrees at the centre.
    Show answerHide answer

    Model answer

    \(\dfrac{40}{360} \times 2\pi \times 9 = 2\pi = 6.28\) cm.

    Mark scheme

    • \(\dfrac{40}{360}\) — M1
    • \(\dfrac{40}{360} \times 2 \times \pi \times 9\) — M1
    • 6.28 — A1
  2. Question 2 Calculator 3 marks

    A sector of a circle has radius 10 cm and angle \(72^\circ\). Work out the area of the sector. Give your answer correct to 3 significant figures.

    Show answerHide answer

    Model answer

    \(\dfrac{72}{360} \times \pi \times 10^2 = 20\pi = 62.8\) cm².

    Mark scheme

    • \(\dfrac{72}{360}\) — M1
    • \(\dfrac{72}{360} \times \pi \times 10^2\) — M1
    • 62.8 — A1
  3. Question 3 Calculator 4 marks

    A sector of a circle has radius 5 cm and angle \(90^\circ\). Work out the perimeter of the sector. Give your answer correct to 3 significant figures.

    Show answerHide answer

    Model answer

    The arc is \(\dfrac{90}{360} \times 2\pi \times 5 = 7.854\) cm. The perimeter is \(7.854 + 5 + 5 = 17.9\) cm.

    Mark scheme

    • Arc length method — M1
    • 7.854 — A1
    • Adding two radii — M1
    • 17.9 — A1
  4. Question 4 Non-calculator 3 marks

    A sector of a circle has radius 12 cm and angle \(150^\circ\). Work out the length of the arc. Give your answer in terms of \(\pi\).

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    Model answer

    \(\dfrac{150}{360} \times 2 \times \pi \times 12 = \dfrac{5}{12} \times 24\pi = 10\pi\) cm.

    Mark scheme

    • \(\dfrac{150}{360}\) or \(\dfrac{5}{12}\) — M1
    • \(\dfrac{5}{12} \times 24\pi\) — M1
    • \(10\pi\) — A1
  5. Question 5 Non-calculator 3 marks

    A sector of a circle has radius 12 cm. The arc length of the sector is \(8\pi\) cm. Work out the angle at the centre of the sector.

    Show answerHide answer

    Model answer

    \(\dfrac{\theta}{360} \times 24\pi = 8\pi\), so \(\dfrac{\theta}{360} = \dfrac{1}{3}\) and \(\theta = 120^\circ\).

    Mark scheme

    • \(\dfrac{\theta}{360} \times 2\pi \times 12 = 8\pi\) — M1
    • \(\dfrac{\theta}{360} = \dfrac{1}{3}\) — M1
    • 120 — A1
  6. Question 6 Calculator 4 marks

    A sector of a circle has radius 8 cm and angle \(90^\circ\). Work out the area of the segment formed by the chord joining the ends of the two radii. Give your answer correct to 3 significant figures.

    Show answerHide answer

    Model answer

    Sector \(= 16\pi = 50.27\) cm². Triangle \(= \dfrac{1}{2} \times 8 \times 8 = 32\) cm². Segment \(= 50.27 - 32 = 18.3\) cm².

    Mark scheme

    • Sector area 50.27 — M1
    • Triangle area 32 — M1
    • \(50.27 - 32\) — M1
    • 18.3 — A1

Quick check

  1. What fraction of a circle is a sector with angle \(90^\circ\)?

    1. A\(\dfrac{1}{2}\)
    2. B\(\dfrac{1}{4}\)
    3. C\(\dfrac{1}{3}\)
    4. D\(\dfrac{1}{8}\)
    Show answerHide answer

    B: \(\dfrac{1}{4}\)

    \(\dfrac{90}{360} = \dfrac{1}{4}\).

  2. What is the arc length of a semicircle of radius 4 cm?

    1. A\(2\pi\)
    2. B\(8\pi\)
    3. C\(4\pi\)
    4. D\(16\pi\)
    Show answerHide answer

    C: \(4\pi\)

    Half of \(2\pi \times 4\) is \(4\pi\).

  3. A sector has radius 6 cm and angle \(60^\circ\). What is its area in terms of \(\pi\)?

    1. A\(6\pi\)
    2. B\(36\pi\)
    3. C\(12\pi\)
    4. D\(2\pi\)
    Show answerHide answer

    A: \(6\pi\)

    \(\dfrac{1}{6} \times 36\pi = 6\pi\).

  4. The perimeter of a sector is...

    1. AThe arc only
    2. BTwo arcs
    3. CThe diameter plus the arc
    4. DThe arc plus two radii
    Show answerHide answer

    D: The arc plus two radii

    The arc plus the two straight radii.

  5. A sector has area \(12\pi\) cm² and radius 6 cm. What is the angle?

    1. A\(60^\circ\)
    2. B\(120^\circ\)
    3. C\(90^\circ\)
    4. D\(240^\circ\)
    Show answerHide answer

    B: \(120^\circ\)

    \(\dfrac{\theta}{360} \times 36\pi = 12\pi\), so \(\dfrac{\theta}{360} = \dfrac{1}{3}\) and \(\theta = 120\).

  6. How do you find the area of a segment?

    1. ASector area plus triangle area
    2. BArc length times radius
    3. CSector area minus triangle area
    4. DHalf the circle area
    Show answerHide answer

    C: Sector area minus triangle area

    A segment is a sector with the triangle removed.

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