GCSE Maths

Sectors of Circles

Edexcel
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Introduction Area of a Sector Arc length of a Sector Solved Examples

Introduction

Circle is very important 2-Dimensional shape in geometry. Sector is a part of circle. Sector’s important characteristics are –

  • A Sector of circle is the portion made by two radii and the arc connecting the ends of those radii. The shape can be viewed as a pizza slice.
Diagram showing a circle divided into two labelled sectors with an arrow pointing from a quarter circle
  • The following are two sectors of same circle
Two sectors of circles with angle theta, radius, and arc labelled in yellow and blue diagrams

Area of a Sector

  • A circle have a complete angle but Sector have a portion of it and we can represent the portion by writing it in fraction.
Circle with 360 degrees, a purple sector with angle theta, and the fraction theta over 360
  • The circle have an area equal to $\pi(r)^2$, So Sector will have area equal to –
$$\text{Area} = \frac{\theta}{360} \times \pi r^2$$

Arc length of a Sector

  • Sector is a part of circle similarly the arc of sector is part of the circumference of whole circle:
Circle with a blue sector showing sector part as theta over 360 and circumference as 2ฯ€r
  • Thus, the formula of Arc length of Sector part is –
$$\text{Arc Length} = \frac{\theta}{360} \times 2\pi r$$

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Solved Examples

Solved Example
Find the area of following sectors:
Three sector diagrams labeled (a), (b), and (c) showing angles of 50, 180, and 90 degrees with different radii
SOLUTION

Formula for finding area of sector:

$$ \text{Area} = \frac{\theta}{360} \times \pi r^2 $$

(a) The area of sector with angle 50ยฐ and radius 5cm:

$$ \text{Area} = \frac{50}{360} \times \pi \times 5^2 \approx 10.91 \text{ cm}^2 $$

(b) The area of sector with angle 180ยฐ and radius 2cm:

$$ \text{Area} = \frac{180}{360} \times \pi \times 2^2 = \frac{1}{2} \times 4\pi \approx 6.28 \text{ cm}^2 $$

(c) The area of sector with angle 90ยฐ and radius 3cm:

$$ \text{Area} = \frac{90}{360} \times \pi \times 3^2 = \frac{1}{4} \times 9\pi \approx 7.065 \text{ cm}^2 $$

Final Answer: (a) Area = $10.91 \text{ cm}^2$, (b) Area = $6.28 \text{ cm}^2$, and (c) Area = $7.065 \text{ cm}^2$

Solved Example
Find out the arc length of the sector given in diagram:
Sectors of Circles with 45 degree angle and 5 cm radius, with coloured arc and segment
SOLUTION

The arc length of sector of circle with radius 5cm and angle 45ยฐ:

$$ \text{Arc Length} = \frac{45}{360} \times 2 \times \pi \times 5 = \frac{1}{8} \times 10\pi \approx 3.93 \text{ cm} $$

Final Answer: Arc Length = $3.93 \text{ cm}$

Solved Example
Work out the arc length of the Sectors:
Sectors of Circles: Two diagrams of sectors labeled (a) and (b), showing 180 and 90 degree angles with inner shaded circles and radii 3 cm and 4 cm
SOLUTION

The formula of finding arc length:

$$ \text{Arc Length} = \frac{\theta}{360} \times 2\pi r $$

(a) Arc length of sector with radius 3cm and angle 180 degree:

$$ \text{Arc Length} = \frac{180}{360} \times 2 \times \pi \times 3 \approx 9.42 \text{ cm} $$

(b) Arc length of sector with radius 4cm and angle 90 degree:

$$ \text{Arc Length} = \frac{90}{360} \times 2 \times \pi \times 4 \approx 6.28 \text{ cm} $$

Final Answer: (a) Arc Length = $9.42 \text{ cm}$ and (b) Arc Length = $6.28 \text{ cm}$

Solved Example
Work out the Length of arc of sector and area of Sector the fan represents:
Sector of a circle shown as an open hand fan with a 175 degree angle and 5 cm radius
SOLUTION

The formula to find the area of sector and Arc Length:

$$ \text{Area} = \frac{\theta}{360} \times \pi r^2 $$
$$ \text{Arc Length} = \frac{\theta}{360} \times 2\pi r $$

The area of the sector shown in diagram:

$$ \text{Area} = \frac{175}{360} \times \pi \times 5^2 \approx 38.151 \text{ cm}^2 $$

The arc length of sector shown in diagram:

$$ \text{Arc Length} = \frac{175}{360} \times 2 \times \pi \times 5 \approx 15.2 \text{ cm} $$

Final Answer: Area of the sector = $38.151 \text{ cm}^2$ and arc length of sector = $15.2 \text{ cm}$

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