Mathematics Form 1-4 : CHAPTER THIRTY FIVE – AREA PART OF A CIRCLE

Specific Objectives

By the end of the topic, the learner should be able to:

1. Find the area of a sector.
2. Find the area of a segment.
3. Find the area of a common region between two circles.

Content

1. Area of a sector
2. Area of a segment
3. Area of common regions between circles.

Introduction

Sector

A sector is an area bounded by two radii and an arc of a circle. A minor sector has a smaller area compared to a major sector, which is the larger part of the circle bounded by the same two radii.

In the image above, the orange part represents the major sector, while the yellow part represents the minor sector.

The Area of a Sector

The area of a sector subtending an angle θ at the centre of the circle is given by the formula:

A = (θ / 360) × πr²

Example

Find the area of a sector of radius 3 cm, if the angle subtended at the centre is θ.

Solution

Using the formula for the area of a sector:

A = (θ / 360) × π × (3)² = (θ / 360) × π × 9

The exact area depends on the value of θ.

Example

The area of the sector of a circle is 38.5 cm². Find the radius of the circle if the angle subtended at the centre is θ.

Solution

From the formula A = (θ / 360) × πr², rearranging to find r:

r = √( (360 × A) / (θ × π) )

Given the area, the radius r is calculated as 7 cm.

Example

The area of a sector of radius 63 cm is 4158 cm². Calculate the angle subtended at the centre of the circle.

Solution

Using the formula A = (θ / 360) × πr², substitute the known values and solve for θ:

4158 = (θ / 360) × π × (63)²

Calculation details lead to the value of θ.

Area of a Segment of a Circle

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

In the figure above, the shaded region is a segment of the circle with centre O and radius r. Given AB = 8 cm, ON = 3 cm, and angle AOB = θ, find the area of the shaded part.

Solution

The area of the segment is calculated as the area of the sector OAPB minus the area of triangle OAB:

Area of segment = area of sector – area of triangle

= 23.19 cm² – 12 cm²

= 11.19 cm²

Area of a Common Region Between Two Intersecting Circles

Find the area of the intersecting circles above if the common chord AB is 9 cm.

Solution

From calculations:

Distances: 6.614 cm and 3.969 cm (details of calculation omitted)

The area between the intersecting circles is the sum of the areas of segments A and B. The area of each segment is given by the area of the sector minus the area of the triangle.

Using trigonometry, sin θ = 0.75. Find the sine inverse of 0.75 to get θ.

Similarly, for the other segment, sin θ = 0.5625. Find the sine inverse of 0.5625 to get θ.

Therefore, the total area of the region between the intersecting circles is the sum of the two segment areas.

End of Topic

Past KCSE Questions on the Topic

1. The figure below shows a circle of radius 9 cm and centre O. Chord AB is 7 cm long. Calculate the area of the shaded region. (4 marks)

2. The figure below shows two intersecting circles with centres P and Q of radius 8 cm and 10 cm respectively. Length AB = 12 cm.

Calculate:

a) APB (2 marks)

b) AQB (2 marks)

c) Area of the shaded region (6 marks)

3.

The diagram above represents a circle centre O of radius 5 cm. The minor arc AB subtends an angle of 120^o at the centre. Find the area of the shaded part. (3 marks)

4. The figure below shows a regular pentagon inscribed in a circle of radius 12 cm, centre O.

Calculate the area of the shaded part. (3 marks)

5. Two circles of radii 13 cm and 16 cm intersect such that they share a common chord of length 20 cm. Calculate the area of the shaded part.

(10 marks)

6. Find the perimeter of the figure below, given AB, BC, and AC are diameters. (4 marks)

7. The figure below shows two intersecting circles. The radius of circle A is 12 cm and that of circle B is 8 cm.

If the angle MBN = 72^o, calculate:

• The size of the angle MAN
• The length of MN
• The area of the shaded region

8.

In the diagram above, two circles, centres A and C and radii 7 cm and 24 cm respectively intersect at B and D. AC = 25 cm.

a) Show that angle ABC = 90^o

b) Calculate:

• i) the size of obtuse angle BAD
• ii) the area of the shaded part (10 marks)

9. The ends of the roof of a workshop are segments of a circle of radius 10 m. The roof is 20 m long. The angle at the centre of the circle is 120^o as shown in the figure below:

(a) Calculate:

• (i) The area of one end of the roof
• (ii) The area of the curved surface of the roof

(b) What would be the cost to the nearest shilling of covering the two ends and the curved surface with galvanized iron sheets costing shs. 310 per square metre?

10. The diagram below, not drawn to scale, is a regular pentagon circumscribed in a circle of radius 10 cm at centre O.

Find:

• (a) The side of the pentagon
• (b) The area of the shaded region

11. Triangle PQR is inscribed in the circle. PQ = 7.8 cm, PR = 6.6 cm, and QR = 5.9 cm. Find:

• (a) The radius of the circle, correct to one decimal place
• (b) The angles of the triangle
• (c) The area of the shaded region