AREAS RELATED TO CIRCLES

AREAS RELATED TO CIRCLES Now with this knowledge, let us try to find some relations (or formulae) to calculate their areas. Let OAPB be a sector of a circle with centre O and radius r (see Fig. . ). Let the degree measure of Ð AOB be q. You know that area of a circle (in fact of a circular region or disc) is pr2. In a way, we can consider this circular region to be a sector forming an angle of ° (i.e., of degree measure ) at the centre O. Now by applying the Unitary Method, we can arrive at the area of the sector OAPB as follows: When degree measure of the angle at the centre is , area of the sector = pr2 So, when the degree measure of the angle at the centre is , area of the sector = r   Therefore, when the degree measure of the angle at the centre is q, area of the sector = r   = r  . Thus, we obtain the following relation (or formula) for area of a sector of a circle: Area of the sector of angle qqqqq =  r   , where r is the radius of the circle and q the angle of the sector in degrees. Now, a natural question arises : Can we find the length of the arc APB corresponding to this sector? Yes. Again, by applying the Unitary Method and taking the whole length of the circle (of angle °) as 2pr, we can obtain the required length of the arc APB as r . So, length of an arc of a sector of angle qqqqq =    r . Fig. . Fig. . Reprint - AREAS RELATED TO CIRCLES Example : Find the area of the segment AYB shown in Fig. . , if radius of the circle is cm and  AOB = °. (Use  = ) Solution : Area of the segment AYB = Area of sector OAYB – Area of  OAB ( ) Now, area of the sector OAYB =    cm2 = cm2 ( ) For finding the area of  OAB, draw OM  AB as shown in Fig. . . Note that OA = OB. Therefore, by RHS congruence,  AMO  BMO. So, M is the mid-point of AB and  AOM =  BOM =   . Let OM = x cm So, from  OMA, OM OA = cos ° or, x = cos ° =       or, x = So, OM = cm Also, AM OA = sin ° = So, AM = cm Therefore, AB = AM = cm = 3cm  Fig. . Fig. . Reprint - AREAS RELATED TO CIRCLES (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 m long instead of m. (Use  = . ) . A brooch is made with silver wire in the form of a circle with diameter mm. The wire is also used in making diameters which divide the circle into equal sectors as shown in Fig. . . Find : (i) the total length of the silver wire required. (ii) the area of each sector of the brooch. . An umbrella has ribs which are equally spaced (see Fig. . ). Assuming umbrella to be a flat circle of radius cm, find the area between the two consecutive ribs of the umbrella. . A car has two wipers which do not overlap. Each wiper has a blade of length cm sweeping through an angle of °. Find the total area cleaned at each sweep of the blades. . To warn ships for underwater rocks, a lighthouse spreads a red coloured light over a sector of angle ° to a distance of . km. Find the area of the sea over which the ships are warned. (Use  = . ) . A round table cover has six equal designs as shown in Fig. . . If the radius of the cover is cm, find the cost of making the designs at the rate of ` . per cm2. (Use = . ) . Tick the correct answer in the following : Area of a sector of angle p (in degrees) of a circle with radius R is (A) R p   (B) R p  (C) R p   (D) R p   Fig. . Fig. . Fig. . Reprint -