Area of Sector and Segment – SS1 Mathematics Lesson Note (Third Term)

A sector of a circle is the region bounded by two radii and the arc between them. Its area is θ/360° × πr², where r is the radius and θ is the angle at the centre.

Behavioural objectives

By the end of the lesson, students should be able to:

1. state the formula for the area of a sector of a circle;
2. calculate the area of a sector from its radius (or diameter) and the angle at the centre;
3. find the area of a sector when the area of the whole circle is given;
4. define a segment of a circle and tell a major segment from a minor segment;
5. state the formula for the area of a segment;
6. calculate the area of a segment.

What is a sector of a circle?

A sector is the part of a circle enclosed by two radii and the arc between them.

• Radius: a straight line from the centre to any point on the circle.
• Arc: a part of the circumference of the circle.
• Angle of the sector (θ): the angle between the two radii at the centre.
• Minor sector: the smaller sector, with θ less than 180°.
• Major sector: the larger sector, with θ greater than 180°.

How do you find the area of a sector?

The area of a sector is the fraction θ/360° of the area of the whole circle.

Area of sector = θ/360° × πr²

• r: the radius of the circle.
• θ: the angle subtended at the centre by the arc XY (the angle of the sector).
• π: a constant, taken as 22/7 or 3.14 unless the question says otherwise.
• πr²: the area of the whole circle.

Steps for finding the area of a sector

1. Write down the radius. If the diameter is given, halve it.
2. Write down the angle θ at the centre.
3. Write the fraction θ/360.
4. Multiply the fraction by πr².
5. Write the answer in square units, such as cm².

Examples

1. Calculate the area of a sector of a circle which subtends an angle of 45° at the centre of the circle, diameter 28 cm (π = 22/7).
2. The area of a circle PQR with centre O is 72 cm². What is the area of sector POQ, if ∠POQ = 40°?

Solutions

Solution to Example 1

1. Diameter d = 2r = 28 cm, so r = 28 ÷ 2 = 14 cm.
2. Area of sector = θ/360° × πr²
3. = 45/360 × 22/7 × 14²
4. = 1/8 × 22/7 × 14 × 14 cm²
5. = 77 cm²

Solution to Example 2

1. Area of the whole circle PQR = πr² = 72 cm².
2. Area of sector POQ = 40/360 × πr²
3. = 40/360 × 72 cm²
4. = 8 cm²

What is a segment of a circle?

A segment of a circle is the area bounded by a chord and an arc of the circle.

• Chord: a straight line joining two points on the circle.
• Minor segment: the smaller part cut off by the chord.
• Major segment: the larger part cut off by the chord.

How do you find the area of a segment?

The area of a minor segment is the area of the sector minus the area of the triangle formed by the two radii and the chord.

Area of segment = area of sector POQ − area of triangle POQ

= θ/360° × πr² − ½r² sin θ

• r: the radius of the circle.
• θ: the angle subtended by the sector at the centre.
• π: a constant, approximately 22/7.
• ½r² sin θ: the area of the triangle, from ½ab sin C with both sides equal to r.
• Major segment: area of the whole circle minus the area of the minor segment.

Worked example

Find the area of the minor segment cut off by a chord that subtends 90° at the centre of a circle of radius 14 cm (π = 22/7).

1. Area of sector = 90/360 × 22/7 × 14 × 14 = 154 cm².
2. Area of triangle = ½ × 14 × 14 × sin 90° = ½ × 196 × 1 = 98 cm².
3. Area of segment = 154 − 98 = 56 cm².

Evaluation

1. Define a segment of a circle.
2. Write the formula for the area of a sector and say what each letter stands for.
3. The area of a circle is 144 cm². Find the area of a sector of this circle that subtends an angle of 90° at the centre.
4. Write the formula for the area of a segment.

Assignment

Answer all the questions below, showing your working.

1. Calculate the area of a sector of a circle of radius 6 cm which subtends an angle of 70° at the centre (π = 22/7).
   • A. 44 cm²
   • B. 22 cm²
   • C. 66 cm²
   • D. 11 cm²
   • E. 16.5 cm²
2. What is the angle subtended at the centre of a sector of a circle of radius 2 cm if the area of the sector is 2.2 cm²? (π = 22/7)
   • A. 120°
   • B. 31½°
   • C. 43°
   • D. 58°
   • E. 63°
3. What is the radius of a sector of a circle which subtends 140° at its centre and has an area of 99 m²?
   • A. 18 m
   • B. 27 m
   • C. 9 m
   • D. 30 m
   • E. 24 m
4. A sector of 80° is removed from a circle of radius 12 cm. What area of the circle is left?
   • A. 253 cm²
   • B. 704 cm²
   • C. 176 cm²
   • D. 125 cm²
   • E. 352 cm²