Sector areas and arc lengths Quiz (Grade 5)

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Sector areas and arc lengths — Notes & Worked Examples Show / Hide

Explanation:

A sector with angle θ° at the centre is the fraction θ360 of a full circle of radius r: Arc Length = θ360 × 2πr and Area of Sector = θ360 × πr².

Be careful when asked for the perimeter of a sector: the curved arc is only part of the boundary, so you must add the two straight radii as well (Perimeter = Arc Length + 2r).

Example 1: Find the area of a sector with radius 9 cm and angle 120°. Give your answer in terms of π and to 1 d.p.

Solution:

Area = 120360 × π × 9²

= 13 × 81π = 27π cm²

Ans: 27π cm² (84.8 cm²)

Example 2: Work out the total perimeter of a sector with radius 8 cm and angle 75° (to 1 d.p.).

Solution:

Arc = 75360 × 2π(8) = 10.47 cm

Add radii: 10.47 + 8 + 8

Answer: 26.5 cm (1 d.p.)

One Page Guide (PDF) Sector areas and arc lengths Worksheets

Question 1 IGCSE Jun 2025 1H Q11

[2 marks]

OABC is a sector of a circle, centre O

angle AOC = 140°
OA = OC = 16 cm
Calculate the area of the sector.
Give your answer correct to 3 significant figures.

cm²

Worked Solution & Answer

(The area of a sector is (angle ÷ 360°) × πr²)
Area = (140 / 360) × π × 16² = (7/18) × 256π = 312.76... = 313 cm² M1 A1

Answer: 313 cm²

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Question 2

[2 marks]

The diagram shows a shaded sector POQ of a circle with centre O and radius 6.2 cm.

The area of the shaded sector is 82.6 cm²
Calculate the size of angle x.
Give your answer correct to 3 significant figures.

Worked Solution & Answer

(The area of a sector is (angle ÷ 360°) × πr²)
(x / 360) × π × 6.2² = 82.6 M1
⇒ x = (82.6 × 360) / (38.44π) = 246.23° ≈ 246°. A1

Answer: 246

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Question 3 AQA Jun 2024 1H Q12

[3 marks]

A sector has radius 12 cm and angle 60°

Work out the length of the arc.
Give your answer in terms of π

Answer cm

Worked Solution & Answer

Arc length = (60 / 360) × 2 × π × 12 = 1/6 × 24π = 4π cm. M2 A1

Answer: Answer 4π cm

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Question 4

[3 marks]

The diagram shows a sector of a circle of radius 18 cm.

The length of the arc is 4π cm.
Work out the value of x.

x =

Worked Solution & Answer

(x / 360) × 2 × π × 18 = 4π M1
⇒ (x / 10) = 4 M1
⇒ x = 40 A1

Answer: x = 40

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Question 5

[3 marks]

OAB is a sector of a circle with centre O and radius 4.7 m.

The sector has a perimeter of 34.3 m.
Find the size of the reflex angle AOB.
Give your answer correct to the nearest degree.

Worked Solution & Answer

Arc length = 34.3 – 2 × 4.7 = 24.9 m M2
(θ / 360) × 2 × π × 4.7 = 24.9 ⇒ θ = (24.9 × 360) / (9.4π) = 303.55... = 304° A1

Answer: 304

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Question 6

[4 marks]

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

The length of the arc AB is 5π cm.
Work out, in terms of π, the area of the sector.
Give your answer in its simplest form.

cm²

Worked Solution & Answer

(The area of a sector is (angle ÷ 360°) × πr²)
Circumference = 2π(6) = 12π cm M2
Fraction of circle = 5π / 12π = 5/12 (angle = 150°) M1
Sector area = (5/12) × π × 6² = 15π cm². A1

Answer: 15π cm²

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Question 7

[4 marks]

OAB is a sector of a circle with centre O and radius 7 cm.

The area of the sector is 40 cm²
Calculate the perimeter of the sector.
Give your answer correct to 3 significant figures.

Worked Solution & Answer

(The area of a sector is (angle ÷ 360°) × πr²)
(θ / 360) × π × 7² = 40 ⇒ θ = (40 × 360) / (49π) = 93.544° M2
Arc length = (93.544 / 360) × 2π × 7 = 80/7 = 11.4285... cm M1
Perimeter = 11.4285 + 7 + 7 = 25.4285... = 25.4 cm A1

Answer: 25.4 cm

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Question 8 AQA Nov 2019 1H Q26

[4 marks]

Here are a circle and a sector of the circle.
They each have radius r.

Work out the size of angle x.
Give your answer in terms of π

circumference of circle = perimeter of sector

Answer degrees

Worked Solution & Answer

(The circumference of a circle is π × diameter = 2πr)
Circumference of circle = 2π r M1
Perimeter of sector = x360 × 2π r + 2r
Equating them: M1
2π r = x360 × 2π r + 2r
Divide through by r:
2π = 2π x360 + 2 M1
2π - 2 = 2π x360
Divide by 2: π - 1 = π x360
x = 360(π - 1)π = 360 - 360π (or 360π - 360π). A1

Answer: Answer 360 - 360π degrees

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Question 9 AQA Nov 2022 3H Q11

[4 marks]

Part of a running track is the arc of a semicircle joined to a straight line.
The semicircle has diameter 45 metres.
The straight line has length 75 metres.

Abby runs once along this part of the track in 18 seconds.
Work out her average speed.
Give your answer to 2 significant figures.

Answer m/s

Worked Solution & Answer

(Average speed = distance ÷ time)
Arc length of semicircle = 1/2 × π × 45 = 22.5π = 70.686 m M2
Total distance = 70.686 + 75 = 145.686 m M1
Average speed = 145.686 ÷ 18 = 8.0936... = 8.1 m/s (2 s.f.). A1

Answer: Answer 8.1 m/s

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Question 10 OCR Nov 2024 5H Q15

[4 marks]

(b) AOB is a sector of a circle, centre O and radius 8 cm.
Angle AOB = x°.
The arc, AB, has length 2π cm.

(b) AOB is a sector of a circle, centre O and radius 8 cm.
Angle AOB = x°.
The arc, AB, has length 2π cm. [4 marks]

Find the area of the sector.
Give your answer in terms of π.

cm²

Worked Solution & Answer

(b)

(The area of a sector is (angle ÷ 360°) × πr²)
Fraction of circle = 2π2π × 8 = 18 M3
Area of sector = 18 × π × 8² = 8π cm² A1

Answer (b): 8π cm²

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Score: 0 / 10 (0 / 33 marks)

Sector areas and arc lengths Worksheets Next: Sim. equ. graphically →

Grade 5 Quizzes

25 Topics

5A - Combining ratios	5B - Direct and inverse proportion
5C - Reverse percentages	5D - Standard form
5E - Speed	5F - Changing the subject
5G - Quadratic expressions	5H - Solving quadratic equations
5J - Drawing quadratic graphs	5K - Cubic/reciprocal graphs
5L - Simultaneous equations	5M - Sector areas and arc lengths
5N - Sim. equ. graphically	5P - Midpoint of a line segment
5Q - Gradient of a line	5R - Equation of a line
5S - Spheres and cones	5T - Similar shapes (lengths)
5U - Trigonometry (SOHCAHTOA)	5V - Exact trig. values
5W - Probability trees	5X - Vectors
5Y - Venn diagrams	5Z - Density
5AA - Pressure