CHAPTER – 12
AREAS RELATED TO CIRCLES
AREA AND PERIMETER OF CIRCLE, QUADRANT, SEMICIRCLE
Area of Circle = r 2 , Perimeter of Circle = Circumference = 2 r
1
Area of Semicircle = r 2 , Perimeter of Semicircle = r 2r
2
1 1
Area of Quadrant = r 2 , Perimeter of Quadrant = r 2r
4 2
IMPORTANT QUESTONS
Find the diameter of the circle whose area is equal to the sum of the areas of the two circles of
diameters 20 cm and 48 cm.
Solution: Here, radius r1 of first circle = 20/2 cm = 10 cm
and radius r2 of the second circle = 48/2 cm = 24 cm
Therefore, sum of their areas = π r12 + π r22 = π (10)2 + π (24)2 = π × 676
Let the radius of the new circle be r cm. Its area = π r2
Therefore, π r2 = π × 676 r2 = 676 r = 26
Thus, radius of the new circle = 26 cm
Hence, diameter of the new circle = 2×26 cm = 52 cm
Questions for Practice
1. The radii of two circles are 19 cm and 9 cm respectively. Find the radius of the circle which has
circumference equal to the sum of the circumferences of the two circles.
2. The radii of two circles are 8 cm and 6 cm respectively. Find the radius of the circle having area
equal to the sum of the areas of the two circles.
3. Find the diameter of the circle whose area is equal to the sum of the areas of the two circles of
diameters 20 cm and 48 cm.
4. The wheels of a car are of diameter 80 cm each. How many complete revolutions does each
wheel make in 10 minutes when the car is travelling at a speed of 66 km per hour?
5. Find the area of a quadrant of a circle whose circumference is 22 cm.
AREAS OF SECTOR AND SEGMENT OF A CIRCLE
Area of the sector of angle θ 0
r 2 , where r is the radius of the circle and θ the angle of the
360
sector in degrees
length of an arc of a sector of angle θ 2 r , where r is the radius
3600
of the circle and θ the angle of the sector in degrees
Area of the segment APB = Area of the sector OAPB – Area of Δ OAB
r 2 – area of Δ OAB
3600
Area of the major sector OAQB = πr2 – Area of the minor sector
OAPB
Area of major segment AQB = πr2 – Area of the minor segment APB
Area of segment of a circle = Area of the corresponding sector – Area of the corresponding
triangle
, IMPORTANT QUESTIONS
Find the area of the sector of a circle with radius 4 cm and of angle 30°. Also, find the area of
the corresponding major sector (Use π = 3.14).
Solution : Here, radius, r = 4 cm, = 300,
2 300 1
We know that Area of sector = 0
r 0
3.14 4 4 3.14 4 4
360 360 12
12.56
4.19cm 2 (approx.)
3
Area of the corresponding major sector
= πr2 – area of sector OAPB
= (3.14 × 16 – 4.19) cm2
= 46.05 cm2 = 46.1 cm2(approx.)
A chord of a circle of radius 10 cm subtends a right angle at the centre. Find the area of the
corresponding : (i) minor segment (ii) major sector. (Use π = 3.14)
Solutions: Here, radius, r = 10 cm, = 900,
900 1
We know that Area of minor sector = 0
r 2
0
3.14 10 10 314 78.5cm 2
360 360 4
1 1
and Area of triangle AOB = b h 10 10 50cm 2
2 2
Area of minor segment = Area of minor sector –
Area of triangle AOB = 78.5 – 50 = 28.5 cm2.
Area of circle = r 2 3.14 10 10 314cm 2
Area of major sector = Area of circle – Area of minor sector
= 314 – 78.5 = 235.5 cm2
Questions for Practice
1. Find the area of a sector of a circle with radius 6 cm if angle of the sector is 60°.
2. The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5
minutes.
3. Area of a sector of a circle of radius 36 cm is 54 π cm2. Find the length of the corresponding arc
of the sector.
4. The wheel of a motor cycle is of radius 35 cm. How many revolutions per minute must the wheel
make so as to keep a speed of 66 km/h?
5. Find the area of the minor segment of a circle of radius 14 cm, when the angle of the
corresponding sector is 60°.
6. A cow is tied with a rope of length 14 m at the corner of a rectangular field of dimensions 20m ×
16m. Find the area of the field in which the cow can graze.
7. A horse is tied to a peg at one corner of a square shaped grass field of side 15 m by means of a 5
m long rope. 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 10 m long instead of 5 m. (Use π = 3.14)
8. A brooch is made with silver wire in the form of a circle with diameter 35 mm. The wire is also
used in making 5 diameters which divide the circle into 10 equal sectors. Find : (i) the total
length of the silver wire required. (ii) the area of each sector of the brooch.
9. In a circle of radius 21 cm, an arc subtends an angle of 60° 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
10. A chord of a circle of radius 15 cm subtends an angle of 60° at the centre. Find the areas of the
corresponding minor and major segments of the circle. (Use π = 3.14 and 3 = 1.73)
AREAS RELATED TO CIRCLES
AREA AND PERIMETER OF CIRCLE, QUADRANT, SEMICIRCLE
Area of Circle = r 2 , Perimeter of Circle = Circumference = 2 r
1
Area of Semicircle = r 2 , Perimeter of Semicircle = r 2r
2
1 1
Area of Quadrant = r 2 , Perimeter of Quadrant = r 2r
4 2
IMPORTANT QUESTONS
Find the diameter of the circle whose area is equal to the sum of the areas of the two circles of
diameters 20 cm and 48 cm.
Solution: Here, radius r1 of first circle = 20/2 cm = 10 cm
and radius r2 of the second circle = 48/2 cm = 24 cm
Therefore, sum of their areas = π r12 + π r22 = π (10)2 + π (24)2 = π × 676
Let the radius of the new circle be r cm. Its area = π r2
Therefore, π r2 = π × 676 r2 = 676 r = 26
Thus, radius of the new circle = 26 cm
Hence, diameter of the new circle = 2×26 cm = 52 cm
Questions for Practice
1. The radii of two circles are 19 cm and 9 cm respectively. Find the radius of the circle which has
circumference equal to the sum of the circumferences of the two circles.
2. The radii of two circles are 8 cm and 6 cm respectively. Find the radius of the circle having area
equal to the sum of the areas of the two circles.
3. Find the diameter of the circle whose area is equal to the sum of the areas of the two circles of
diameters 20 cm and 48 cm.
4. The wheels of a car are of diameter 80 cm each. How many complete revolutions does each
wheel make in 10 minutes when the car is travelling at a speed of 66 km per hour?
5. Find the area of a quadrant of a circle whose circumference is 22 cm.
AREAS OF SECTOR AND SEGMENT OF A CIRCLE
Area of the sector of angle θ 0
r 2 , where r is the radius of the circle and θ the angle of the
360
sector in degrees
length of an arc of a sector of angle θ 2 r , where r is the radius
3600
of the circle and θ the angle of the sector in degrees
Area of the segment APB = Area of the sector OAPB – Area of Δ OAB
r 2 – area of Δ OAB
3600
Area of the major sector OAQB = πr2 – Area of the minor sector
OAPB
Area of major segment AQB = πr2 – Area of the minor segment APB
Area of segment of a circle = Area of the corresponding sector – Area of the corresponding
triangle
, IMPORTANT QUESTIONS
Find the area of the sector of a circle with radius 4 cm and of angle 30°. Also, find the area of
the corresponding major sector (Use π = 3.14).
Solution : Here, radius, r = 4 cm, = 300,
2 300 1
We know that Area of sector = 0
r 0
3.14 4 4 3.14 4 4
360 360 12
12.56
4.19cm 2 (approx.)
3
Area of the corresponding major sector
= πr2 – area of sector OAPB
= (3.14 × 16 – 4.19) cm2
= 46.05 cm2 = 46.1 cm2(approx.)
A chord of a circle of radius 10 cm subtends a right angle at the centre. Find the area of the
corresponding : (i) minor segment (ii) major sector. (Use π = 3.14)
Solutions: Here, radius, r = 10 cm, = 900,
900 1
We know that Area of minor sector = 0
r 2
0
3.14 10 10 314 78.5cm 2
360 360 4
1 1
and Area of triangle AOB = b h 10 10 50cm 2
2 2
Area of minor segment = Area of minor sector –
Area of triangle AOB = 78.5 – 50 = 28.5 cm2.
Area of circle = r 2 3.14 10 10 314cm 2
Area of major sector = Area of circle – Area of minor sector
= 314 – 78.5 = 235.5 cm2
Questions for Practice
1. Find the area of a sector of a circle with radius 6 cm if angle of the sector is 60°.
2. The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5
minutes.
3. Area of a sector of a circle of radius 36 cm is 54 π cm2. Find the length of the corresponding arc
of the sector.
4. The wheel of a motor cycle is of radius 35 cm. How many revolutions per minute must the wheel
make so as to keep a speed of 66 km/h?
5. Find the area of the minor segment of a circle of radius 14 cm, when the angle of the
corresponding sector is 60°.
6. A cow is tied with a rope of length 14 m at the corner of a rectangular field of dimensions 20m ×
16m. Find the area of the field in which the cow can graze.
7. A horse is tied to a peg at one corner of a square shaped grass field of side 15 m by means of a 5
m long rope. 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 10 m long instead of 5 m. (Use π = 3.14)
8. A brooch is made with silver wire in the form of a circle with diameter 35 mm. The wire is also
used in making 5 diameters which divide the circle into 10 equal sectors. Find : (i) the total
length of the silver wire required. (ii) the area of each sector of the brooch.
9. In a circle of radius 21 cm, an arc subtends an angle of 60° 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
10. A chord of a circle of radius 15 cm subtends an angle of 60° at the centre. Find the areas of the
corresponding minor and major segments of the circle. (Use π = 3.14 and 3 = 1.73)
