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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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Student handouts

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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 \(\pi\)?

    Show answerHide answer

    \(36\pi\)

  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
\(\dfrac{\theta}{360}\).

Questions and answers

12 questions set on this lesson, with the mark schemes and model answers open.

1. Exam question Calculator 3 marks Easier

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.

Mark scheme — 3 marks available

  • \(\dfrac{40}{360}\) — M1
  • \(\dfrac{40}{360} \times 2 \times \pi \times 9\) — M1
  • 6.28 — A1

Model answer

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

2. Exam question Calculator 3 marks Easier

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.

Mark scheme — 3 marks available

  • \(\dfrac{72}{360}\) — M1
  • \(\dfrac{72}{360} \times \pi \times 10^2\) — M1
  • 62.8 — A1

Model answer

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

3. Exam question Calculator 4 marks Easier

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.

Mark scheme — 4 marks available

  • Arc length method — M1
  • 7.854 — A1
  • Adding two radii — M1
  • 17.9 — A1

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.

4. Exam question Non-calculator 3 marks Easier

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 \(\pi\).

Mark scheme — 3 marks available

  • \(\dfrac{150}{360}\) or \(\dfrac{5}{12}\) — M1
  • \(\dfrac{5}{12} \times 24\pi\) — M1
  • \(10\pi\) — A1

Model answer

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

5. Exam question Non-calculator 3 marks Easier

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.

Mark scheme — 3 marks available

  • \(\dfrac{\theta}{360} \times 2\pi \times 12 = 8\pi\) — M1
  • \(\dfrac{\theta}{360} = \dfrac{1}{3}\) — M1
  • 120 — A1

Model answer

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

6. Exam question Calculator 4 marks Easier

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.

Mark scheme — 4 marks available

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

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.

7. Multiple choice 1 mark Easier

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

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

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

8. Multiple choice 1 mark Core

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

  1. A \(2\pi\)
  2. B \(8\pi\)
  3. C \(4\pi\) Correct
  4. D \(16\pi\)

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

9. Multiple choice 1 mark Core

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

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

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

10. Multiple choice 1 mark Core

The perimeter of a sector is...

  1. A The arc only
  2. B Two arcs
  3. C The diameter plus the arc
  4. D The arc plus two radii Correct

Why: The arc plus the two straight radii.

11. Multiple choice 1 mark Core

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

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

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

12. Multiple choice 1 mark Stretch

How do you find the area of a segment?

  1. A Sector area plus triangle area
  2. B Arc length times radius
  3. C Sector area minus triangle area Correct
  4. D Half the circle area

Why: A segment is a sector with the triangle removed.