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3. Find the area of the shaded region in Fig, if ABCD is a square of side 14 cm and APD and BPC are semicircles.

                    

Area of the shaded region is given by =   Area of the square - Area of two semicircles. Area of square is    :     And the area of the semicircle is:-                                                                                                                                                                             Hence the area of the shaded region is given by :   

14.    Tick the correct answer in the following :

Area of a sector of angle p (in degrees) of a circle with radius R is

                (A)    \frac{p}{180}\times 2\pi R

                (B)    \frac{p}{180}\times \pi R^2

                (C)    \frac{p}{360}\times 2\pi R

                (D)    \frac{p}{720}\times 2\pi R^2

We know that the area of the sector is given by:-                                                                                                                     Hence option (d) is correct.

13.  A round table cover has six equal designs as shown in Fig. . If the radius of the cover is 28 cm, find the cost of making the designs at the rate of Rs. 0.35 per cm2. (Use \sqrt3 = 1.7)

                                          

The angle of each of the six sectors is 60o at the centre.                                                        Area of the sector is given by:-                                                         or                                                               or                                                               And the area of the equilateral triangle associated with...

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 π = 3.14)

The area of sector is given by :-                                          In this case the angle is 80o. Thus the area is :-                                                        or                                                 

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.

The area cleaned by one wiper is:-                                                      or                                                              or                                                             Hence the required area (area cleaned by both blades) is given by:-                                                                

10.  An umbrella has 8 ribs which are equally spaced (see Fig. ). Assuming umbrella to be a flat circle of radius 45 cm, find the area between the two consecutive ribs of the umbrella.

                    

It is given that the umbrella has 8 ribs so the angle of each sector is 45o. Thus the area of the sector is given by:-                                                                                                 Hence the area between two consecutive ribs is       .

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 :

 (ii) the area of each sector of the brooch.

                        

The total number of lines present in the brooch are 10 (line starting from centre). Thus the angle of each sector is 36o. The area of sector is given by :-                                                                                                                                  

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.  Find :

 (i) the total length of the silver wire required.

                        

The total wire required will be for 5 diameters and the circumference of the brooch. The circumference of the brooch:-                                                              Hence the total wire required will be:-      .

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.). Find

(ii) the increase in the grazing area if the rope were 10 m long instead of 5 m. (Use \pi = 3.14)

                              

When the length of the rope is 10 m, the area grazed will be:-                                                                                                                                                                            Hence the change in grazing area is given by  :                                                                       

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.). Find

  (i) the area of that part of the field in which the horse can graze.

                                      

The part grazed by the horse is given by     =     Area of sector                                                                                                                                                                                                                                                                           

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 \pi = 3.14 and \sqrt3 = 1.73)

For the area of the segment, we need the area of sector and area of the associated triangle. So, the area of the sector is :                                             or                                         Now, consider the triangle:-             Draw a perpendicular from the centre of the circle on the base of the triangle (let it be h). Using geometry we can write,                      ...

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 \pi = 3.14 and \sqrt 3 = 1.73)

The area of the sector is :                                                                                                                                             Now consider the triangle, the angle of the sector is 600. This implies it is an equilateral triangle. (As two sides are equal so will have the same angle. This possible only when all angles are equal i.e., 60o.)  Thus, the area...

5. In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find:

  (iii) area of the segment formed by the corresponding chord

For area of segment we need to subtract area of the triangle attached with the area of arc. Thus consider the triangle :- It is given that angle of arc is 60o, or we can say that all angles are 60o (since two sides are equal). Hence it is an equilateral triangle.   Area of triangle is :-                                            Hence the area of segment is :-                                  ...

5. In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find:

 (ii) area of the sector formed by the arc

We know that the area of the sector is given by:-                                                                                                                    Thus the area of the sector is 231 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

The length of the arc is given by:-                                                                                                                                                             Hence the length of the arc is 22 cm.

4. A chord of a circle of radius 10 cm subtends a right angle at the centre. Find the area of the corresponding :

 (ii) major sector. (Use π = 3.14)

The area of the major sector can be found directly by using the formula :                                                         In the case of this, the angle is  360o  -  90o  =  270o. Thus the area is : -                                                                     

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

The angle in the minor sector is 90o. Thus the area of the sector is given by :-                                                                                                           Now area of triangle is  :-                                                              Thus area of minor segment   =   Area of sector    -   Area of triangle or                                           

3. The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes.

The minute hand rotates 360o in one hour. We need to find rotation in 5 min.  :-                                                                        The area of sector is given by :                                                                                                                          Hence the area swept by minute hand in 5 minutes is .  

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

We are given the circumference of circle. Thus,                                      Also, we know that the area of a sector is given by :                                                                             It is given that we need to find area of a quadrant thus   Hence area becomes :-                                                                                                    ...

1. Find the area of a sector of a circle with radius 6 cm if angle of the sector is 60°.

We know that the area of a sector having radius r and angle  is given by:-                                                      Thus the area of the given sector is:-                                                                                                                        
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