GCSE Maths revision · Geometry and measures

Arcs and sectors

Foundation and Higher

The idea

A sector is a fraction $\frac{\theta}{360}$ of a whole circle. So arc length $= \frac{\theta}{360} \times 2\pi r$ and sector area $= \frac{\theta}{360} \times \pi r^2$ .

Write the fraction of the circle: $\frac{\theta}{360}$ .
Multiply by the full circumference for an arc, or the full area for a sector.
For the perimeter of a sector, add the two radii to the arc.

Watch out The arc length is only the curved part; the perimeter adds two radii.

See it worked

Worked exampleCalculator allowed2 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 1 decimal place.

The sector is $\frac{72}{360} = \frac{1}{5}$ of the circle. $\tfrac{72}{360} \times \pi \times 10^2 = 20\pi = 62.83\ldots$

Answer $62.8\text{ cm}^2$

Your turn

Question 1Calculator allowed3 marks

The diagram shows a round pizza with a diameter of 30 cm. It is cut into 8 equal slices from the centre.

Work out the area of one slice.

Give your answer correct to 3 significant figures.

Question 2Non-calculator4 marks

The diagram shows a sector of a circle with angle $45^\circ$ .

The area of the sector is $8\pi$ cm².

(a) Work out the value of $r$ .

(b) Work out the exact arc length of the sector.

Question 3Non-calculator6 marks

The diagram shows a sector $OAB$ of a circle with centre $O$ and radius 6 cm.

The area of the sector is $15\pi\text{ cm}^2$ .

(a) Work out the angle $AOB$ .

(b) Work out the perimeter of the sector. Give your answer in the form $a + b\pi$ .

(c) The sector is folded so that $OA$ meets $OB$ , making a cone with no base. Work out the radius of the base of the cone.

In past papers

61 official questions in 61 papers since 2017. The latest:

All GCSE Maths past papers

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