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Areas Related to Circles — Class 10 Mathematics

Mathematics · Chapter 11

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Chapter summary

This chapter computes lengths and areas associated with circles and their parts. Building on the circumference and area of a circle, it introduces the sector — a “slice” bounded by two radii and an arc — and the segment, bounded by a chord and an arc. Formulae for the length of an arc and the area of a sector in terms of the central angle let students find areas of these regions, and combined figures are handled by adding or subtracting the relevant areas. The chapter connects geometry with measurement in everyday shapes.

Perimeter and area of a circleArea of a sectorArea of a segmentAreas of combinations of figures

Key ideas & terms

Circumference: The perimeter of a circle, .
Sector: The region between two radii and the arc between them.
Segment: The region between a chord and its arc.
Area of a sector: , where is the central angle.

Extra questions & answers

Write the formula for the area of a sector of angle .

, where is the radius.

What is the difference between a sector and a segment?

A sector is bounded by two radii and an arc; a segment is bounded by a chord and an arc.

Write the formulae for the circumference and area of a circle of radius r.

Circumference and area .

What is a sector of a circle?

The region enclosed by two radii of the circle and the arc between them (refer to the figure in your NCERT textbook).

Write the formula for the area of a sector of angle .

Area of sector .

Write the formula for the length of an arc of angle .

Length of arc .

What is a segment of a circle?

The region enclosed by a chord and the arc corresponding to it.

Find the area of a sector of radius 7 cm and angle . (Take .)

Area cm.

How do you find the area of the major sector if the minor sector’s area is known?

Subtract the area of the minor sector from the total area of the circle .

How is the area of a combination of plane figures found?

By adding or subtracting the areas of the basic figures (circles, sectors, triangles, rectangles) that make up the combined shape.

EXERCISE 11.1

1. Find the area of a sector of a circle with radius if angle of the sector is .

The area of the sector is

Therefore, the required area is , or .

2. Find the area of a quadrant of a circle whose circumference is .

Let the radius be . Since the circumference is ,

Using ,

The area of the quadrant is

Therefore, the required area is , or .

3. The length of the minute hand of a clock is . Find the area swept by the minute hand in minutes.

In minutes, the minute hand turns through . Therefore, in minutes, it turns through

The area swept is

Therefore, the area swept is , or .

4. A chord of a circle of radius subtends a right angle at the centre. Find the area of the corresponding : (i) minor segment (ii) major sector. (Use )

The angle at the centre is . The area of the minor sector is

The triangle formed by the two radii is right-angled, so its area is

Thus, the area of the minor segment is

The major sector has angle . Its area is

Therefore, (i) the minor segment has area , and (ii) the major sector has area .

5. In a circle of radius , an arc subtends an angle of at the centre. Find:
(i) the length of the arc (ii) area of the sector formed by the arc
(iii) area of the segment formed by the corresponding chord

(i) The length of the arc is

(ii) The area of the sector is

(iii) Since the two radii are and the included angle is , the triangle is equilateral. Its area is

Therefore, the area of the segment is

6. A chord of a circle of radius subtends an angle of at the centre. Find the areas of the corresponding minor and major segments of the circle.
(Use and )

The area of the sector is

The triangle formed by the two radii is equilateral. Its area is

Thus, the area of the minor segment is

The area of the whole circle is

Therefore, the area of the major segment is

7. A chord of a circle of radius subtends an angle of at the centre. Find the area of the corresponding segment of the circle.
(Use and )

The area of the sector is

The area of the triangle formed by the radii is

Since ,

Therefore, the area of the corresponding segment is

8. A horse is tied to a peg at one corner of a square shaped grass field of side by means of a long rope (see Fig. 11.8). Find
(i) the area of that part of the field in which the horse can graze.
(ii) the increase in the grazing area if the rope were long instead of . (Use )

The horse can graze over a quadrant because it is tied at a corner of the square.

(i) With a rope, the grazing area is

(ii) With a rope, the grazing area is

Hence, the increase in grazing area is

Therefore, (i) the grazing area is , and (ii) the increase is .

9. A brooch is made with silver wire in the form of a circle with diameter . The wire is also used in making diameters which divide the circle into equal sectors as shown in Fig. 11.9. Find :
(i) the total length of the silver wire required.
(ii) the area of each sector of the brooch.

(i) The length of wire used for the circular boundary is

The length used for diameters is

Thus, the total length of wire is

(ii) The radius is . The area of the circle is

Since there are equal sectors, the area of each sector is

Therefore, (i) of wire is required, and (ii) each sector has area .

10. An umbrella has ribs which are equally spaced (see Fig. 11.10). Assuming umbrella to be a flat circle of radius , find the area between the two consecutive ribs of the umbrella.

The equally spaced ribs divide the umbrella into equal sectors. Therefore, the angle between two consecutive ribs is

The required area is

Therefore, the area between two consecutive ribs is approximately .

11. A car has two wipers which do not overlap. Each wiper has a blade of length sweeping through an angle of . Find the total area cleaned at each sweep of the blades.

The area cleaned by one wiper is

Since the two swept regions do not overlap, the total area cleaned is

Therefore, the total area cleaned is approximately .

12. To warn ships for underwater rocks, a lighthouse spreads a red coloured light over a sector of angle to a distance of . Find the area of the sea over which the ships are warned.
(Use )

The warned region is a sector of radius and angle . Its area is

Since ,

Therefore, the area of the sea over which ships are warned is .

13. A round table cover has six equal designs as shown in Fig. 11.11. If the radius of the cover is , find the cost of making the designs at the rate of ₹ per . (Use )

The six equal designs together occupy the area between the circle and the inscribed regular hexagon.

The area of the circular cover is

The regular hexagon consists of equilateral triangles of side . Its area is

Thus, the total area of the designs is

The cost of making the designs is

Therefore, the required cost is ₹ .

14. Tick the correct answer in the following :
Area of a sector of angle (in degrees) of a circle with radius is
(A) (B) (C) (D)

The area of a sector of angle is

Option (D) simplifies to

Therefore, the correct answer is (D).

Areas Related to Circles — Class 10 Mathematics

CBSE Class 10 community

Chapter summary

Key ideas & terms

Extra questions & answers

Write the formula for the area of a sector of angle .

What is the difference between a sector and a segment?

Write the formulae for the circumference and area of a circle of radius r.

What is a sector of a circle?

Write the formula for the area of a sector of angle .

Write the formula for the length of an arc of angle .

What is a segment of a circle?

Find the area of a sector of radius 7 cm and angle . (Take .)

How do you find the area of the major sector if the minor sector’s area is known?

How is the area of a combination of plane figures found?

EXERCISE 11.1

1. Find the area of a sector of a circle with radius if angle of the sector is .

2. Find the area of a quadrant of a circle whose circumference is .

3. The length of the minute hand of a clock is . Find the area swept by the minute hand in minutes.

4. A chord of a circle of radius subtends a right angle at the centre. Find the area of the corresponding : (i) minor segment (ii) major sector. (Use )

5. In a circle of radius , an arc subtends an angle of at the centre. Find:(i) the length of the arc (ii) area of the sector formed by the arc(iii) area of the segment formed by the corresponding chord

6. A chord of a circle of radius subtends an angle of at the centre. Find the areas of the corresponding minor and major segments of the circle.(Use and )

7. A chord of a circle of radius subtends an angle of at the centre. Find the area of the corresponding segment of the circle.(Use and )

8. A horse is tied to a peg at one corner of a square shaped grass field of side by means of a long rope (see Fig. 11.8). Find(i) the area of that part of the field in which the horse can graze.(ii) the increase in the grazing area if the rope were long instead of . (Use )

9. A brooch is made with silver wire in the form of a circle with diameter . The wire is also used in making diameters which divide the circle into equal sectors as shown in Fig. 11.9. Find :(i) the total length of the silver wire required.(ii) the area of each sector of the brooch.

10. An umbrella has ribs which are equally spaced (see Fig. 11.10). Assuming umbrella to be a flat circle of radius , find the area between the two consecutive ribs of the umbrella.

11. A car has two wipers which do not overlap. Each wiper has a blade of length sweeping through an angle of . Find the total area cleaned at each sweep of the blades.

12. To warn ships for underwater rocks, a lighthouse spreads a red coloured light over a sector of angle to a distance of . Find the area of the sea over which the ships are warned.(Use )

13. A round table cover has six equal designs as shown in Fig. 11.11. If the radius of the cover is , find the cost of making the designs at the rate of ₹ per . (Use )

14. Tick the correct answer in the following :Area of a sector of angle (in degrees) of a circle with radius is(A) (B) (C) (D)

5. In a circle of radius , an arc subtends an angle of at the centre. Find:
(i) the length of the arc (ii) area of the sector formed by the arc
(iii) area of the segment formed by the corresponding chord

6. A chord of a circle of radius subtends an angle of at the centre. Find the areas of the corresponding minor and major segments of the circle.
(Use and )

7. A chord of a circle of radius subtends an angle of at the centre. Find the area of the corresponding segment of the circle.
(Use and )

8. A horse is tied to a peg at one corner of a square shaped grass field of side by means of a long rope (see Fig. 11.8). Find
(i) the area of that part of the field in which the horse can graze.
(ii) the increase in the grazing area if the rope were long instead of . (Use )

9. A brooch is made with silver wire in the form of a circle with diameter . The wire is also used in making diameters which divide the circle into equal sectors as shown in Fig. 11.9. Find :
(i) the total length of the silver wire required.
(ii) the area of each sector of the brooch.

12. To warn ships for underwater rocks, a lighthouse spreads a red coloured light over a sector of angle to a distance of . Find the area of the sea over which the ships are warned.
(Use )

14. Tick the correct answer in the following :
Area of a sector of angle (in degrees) of a circle with radius is
(A) (B) (C) (D)