12. AREAS RELATED TO CIRCLES (NCERT)
NCERT
Chapter 12. AREAS RELATED TO CIRCLES
Chapter 12 . Areas related to circles |
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Exercise 12.1 solution complete Exercise 12.2 solution complete Exercise 12.3 solution complete |
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1. Circumference of a circle 3. The sector of a circle :
Here, OAPB is the minor sector of the circle and OAQB is the major sector of the circle . 4. The segment of a circle :
Here, APB is the minor segment of the circle and AQB is the major segment of the circle . OR Area of the segment APB = Area of the sector OAPB – Area of ∆ OAB = 8. In figure :
Area of the major sector OAQB = In figure : Area of major segment AQB = |
.
.
⋅
– Area of the minor sector OAPB and
EXERCISE 12.1
Unless stated otherwise , use 
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 .
Solution: Let 
be the radius of new circle .
Here, 
cm and
cm .
A/Q , 
cm
cm
Therefore , the radius of the new circle is 28 cm .
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 .
Solution: Let be the radius of new circle .
Here, 
cm and 
cm .
A/Q, 
cm
Therefore, the radius of the new circle is 10 cm .
3. Fig. 12.3 depicts an archery target marked with its five scoring regions from the centre outwards as Gold , Red , Blue , Black and White . The diameter of the region representing Gold score is 21 cm and each of the
other bands is 10.5 cm wide . Find the area of each of the five scoring regions .
Solution: For Gold region : Here , 
cm , 
cm
Area of the gold region 
For Red region : Here , 
Area of the red region 
For blue region : Here ,
Area of the red region
For Black region : Here ,
Area of the black region
For White region :
Here , 
Area of the white region
4. The wheels of a car are of diameter 80 cm each . How many complete revolution does each wheel make in 10 minutes when the car is travelling at a speed of 66 km per hour ?
Solution: Here, 
and 
The perimeter of the wheels of a car 
The speed of the car 
The distances travel by the car 
Speed of the car × time 
[Time = 10 minutes]
The number of complete revolution of the wheels of the car 
5. Tick the correct answer in the following and justify your choice : If the perimeter and the area of a circle are numerically equal, then the radius of the circle is :
(A) 2 units (B) 
units (C) 4 units (D) 7 units
Solution: Let 
be the radius of the circle .
A/Q , 
or 
units
(A) 2 units
EXERCISE 12.2
Unless stated otherwise , use .
1. Find the area of a sector of a circle with radius 6 cm if angle of the sector is 60° .
Solution : Here, 
and 
The area of a sector of a circle 
2. Find the area of a quadrant of a circle whose circumference is 22 cm .
Solution : We have, 
m
The area of a quadrant of a circle 
3. The length of the minute hand of a clock is 14 cm . Find the area swept by the minute hand in 5 minutes .
Solution :
4. 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 
)
Solution : Here, 
The area of the minor segment of a circle 
78.5 cm²
5. 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 .
Solution: Here, Radius 
cm and
(i) the length of an arc of a sector 
× 
× 
× 
× 
× 2 × 22 × 3 22 cm
(ii) the area of the sector formed by the arc × 
× × 21 × 21 × 22 × 3 ×21 
cm2
(iii) Here, OA = OB = 21 cm and 
× 
× 120° 60°
So, OAB be an equilateral triangle .
Area of triangle OAB 
× (Side)² × 21 × 21 
cm²
The area of segment formed by chord Area of sector – Area of triangle 231 - 
× 
cm²
(6) 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 
and 
)
Solution: Here , radius 
cm , 
Area of the minor segment = Area of the sector – Area of the triangle formed by radius and chord
× 3.14 × 15 × 15 – × 15 × 15 × 
cm²
Area of major segment = Area of the circle – Area of the minor segment 
cm² 
cm²
7. A chord of a circle of radius 12 cm subtends an angle of 120° at the centre . Find the area of the corresponding segment of the circle . (Use and )
Solution :
8. 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 (see Fig. 12.11) . 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 )
Solution :
9. 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 as shown in Fig. 12.12 . Find :
(i) the total length of the silver wire required . (ii) the area of each sector of the brooch .
Solution: (i) Here, diameter 
mm and Radius 
352
mm
The circumference = 
2 × × 
mm
The length of the wire required to make 5 diameters 
mm
Therefore, the total length of the silver wire required 
mm 
mm
(ii) The angle of each brooch 
36°
So, the area of each sector of the brooch 
× 
× × 
× 
= 11 ×
= 
cm²
10 . An umbrella has 8 ribs which are equally spaced (see Fig. 12.13) . Assuming umbrella to be a flat circle of radius 45 cm, find the area between the two consecutive ribs of the umbrella .
Solution : Here , 
cm
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 25 cm sweeping through an angle of 115° . Find the total area cleaned at each sweep of the blades .
Solution : Here , 
The area cleaned at each sweep of the blades 
Therefore, 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 80° to a distance of 16.5 km . Find the area of the sea over which the ships are warned . (Use ) .
Solution: Here , 
The area of the sea over which the ships are warn 
13. A round table cover has six equal designs as shown in Fig. 12.14 . If the radius of the cover is 28 cm , find the cost of making the designs at the rate of Rs 0.35 per 
. (Use )
Solution :
14.Tick the correct answer in the following :
Area of a sector of angle 
(in degrees) of a circle with radius R is :
(A) 
(B) 
(C) 
(D) 
Solution : (D)
EXERCISE 12.3
Unless stated otherwise , use .
1. Find the area of the shaded region in Fig.12.19, if 
and O is the centre of the circle .
Solution: Here , 
cm and 
cm
we have ,
Radius 
cm
Area of 
cm²
Area of semicircle 
cm²
The area of the shaded region Area of semicircle 
Area of 
cm²
2. Find the area of the shaded region in Fig. 12.20 , if radii of the two concentric circles with centre O are 7 cm and 14 cm respectively and ∠AOC=40° .
Solution:
3. Find the area of the shaded region in Fig. 12.21 , if ABCD is a square of side 14 cm and APD and BPC are semicircles .
Solution:
4. Find the area of the shaded region in Fig. 12.22 , where a circular arc of radius 6 cm has been draw with vertex O of an equilateral triangle OAB of side 12 cm as centre .
Fig. 12.22
Solution:
5. From each corner of a square of side 4 cm a quadrant of a circle of radius 1 cm is cut and also a circle of diameter 2 cm is cut as shown in Fig. 12.23 . Find the area of the remaining portion of the square .
Solution:
6. In a circular table cover of radius 32 cm , a design is formed leaving an equilateral triangle ABC in the middle as shown in Fig. 12.24 . Find the area of the design .
Solution:
7. In Fig. 12.25, ABCD is a square of side 14 cm . With centres A , B , C and D , four circles are drawn such that each circle touch externally two of the remaining three circles . Find the area of the shaded region .
Solution:
8.In Fig. 12.36 depicts a racing track whose left and right ends are semicircular . The distance between the two inner parallel line segments is 60 m and they are each 106 m long . If the track is 10 m wide , find :
(i) the distance around the track along its inner edge .
(ii) the area of the track .
Solution:
9. In Fig. 12.27, AB and CD are two diameters of a circle (with centre O) perpendicular to each other and OD is the diameter of the smaller circle . If 
cm , find the area of the shaded region .
Solution:
10. The area of an equilateral triangle ABC is 17320.5 cm² . With each vertex of the triangle as centre , a circle is drawn with radius equal to half the length of the side of the triangle (see Fig. 12.28) . Find the area of the shaded region . (Use 
)
Solution:
11. On a square handkerchief , nine circular designs each of radius 7 cm are made (see Fig. 12.29) . Find the area of the remaining portion of the handkerchief .
Solution:
12. In Fig. 12.30 , OACB is a quadrant of a circle with centre O and radius 3.5 cm . If 
cm , find the area of the (i) quadrant OACB , (ii) shaded region.
Solution:
13. In Fig. 12.31, a square OABC is inscribed in a quadrant OPBQ . If 
cm, find the area of the shaded region . (Use )
Solution: Given, 
be a square .
So , 
cm
Area of square 
cm²
In 
, we have 
cm
Radius 
cm
Area of the quadrant 
cm²
Therefore, the area of shaded region 
cm²
14. AB and CD are respectively arcs of two concentric circles of radii 21cm and 7 cm and centre O (see Fig. 12.32) . If 
, find the area of the shaded region.
Solution:
15. In Fig. 12.33 , ABC is a quadrant of a circle of radius 14 cm and a semicircle is drawn with BC as diameter . Find the area of the shaded region .
Solution:
16. Calculate the area of the designed region in Fig. 12.34 common between the two quadrants of circles of radius 8 cm each .
Solution:
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