Chapter 12. AREAS RELATED TO CIRCLES

Chapter 12 . Areas related to circles

Exercise 12.1 solution complete

1. Circumference of a circle . 2. Area 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 . 5. Length of an arc of a sector of a circle with radius r and angle with degree measure θ is  ⋅ 6. Area of a sector of a circle with radius r and angle with degree measure θ is 7. Area of segment of a circle = Area of the corresponding sector – Area of the corresponding triangle . OR Area of the segment APB = Area of the sector OAPB – Area of ∆ OAB =    – area of OAB  8. In figure :  Area of the major sector OAQB =  – Area of the minor sector OAPB and   In figure :    Area of major segment AQB =  – Area of the minor segment APB .

Class 10 Maths Chapter 12. AREAS RELATED TO CIRCLES Exercise 12.1 Solutions

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, cm and

        

 The area of a sector AOB 

2. Find the area of a quadrant of a circle whose circumference is 22 cm .

Solution : We have, 

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:   Here,

The angle of swept in 5 minutes

Therefore, the area swept by the minute hand in 5 minutes

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 ACB

(ii)  Here,OA = OB = and

    

Area of major sector OADB

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 ACB

(ii)  The area of the sector OACB

(iii)  Here, OA = OB = 21 cm  and

So, OAB be an equilateral  triangle .

     Area of triangle OAB

The area of the segment ACB = Area of sector  – Area of triangle

(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 ACB =  Area of the sector OACB – 

 

Area of major segment of a circle = Area of the circle – Area of the minor segment of a circle 

 

  

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: Here,  and

         

The area of the segment ACB

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:   Since , ABCD is a square .

So, AB = BC = CD = AD = 15 cm

     

(i)  Here, and

Therefore, the area of the field in which the horse can graze

 

(ii) Here,  ,  and

The increasing area of the field

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   and Radius

The circumference

 

The length of the wire required to make 5 diameters  

Therefore, the total length of the silver wire required   

(ii)  The angle of each brooch

So, the area of each sector of the brooch

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:  Here, and

 

Area of a design part of the round table cover

Area of six design part of the round table cover

Therefore, the cost of making the designs

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)