6 Are By studying this lesson you will be ble to find the res of sectors of circles, solve problems relted to the res of compound plne figures contining sectors of circles. Ares of plne figures Let us recll some fcts you hve lernt in previous grdes under the topic Are. Nme Plne Figure How the re is clculted Formul for the re (A) Rectngle b length «bredth A = «b Squre (length of side) 2 A = 2 Prllelogrm h bse «ltitude A = «h Tringle h «bse «ltitude Trpezium b h «sum of the lengths of the prllel sides «ltitude Circle r π «(rdius) 2 A = πr 2 65
Review Exercise. Find the re of ech of the following plne figures. 9 cm 4 cm 4 cm A B C D 8 cm 24 cm E 2 cm 0 cm F G 2. The rectngle in Figure C hs been formed by joining together the trpezium in Figure A nd the tringle in Figure B. Figure A Figure B Figure C 5 cm 3 cm 4 cm 5 cm 4 cm 8 cm (i) Find the re of the trpezium in Figure A. (ii) Find the re of the tringle in Figure B. (iii) Find the re of the rectngle in Figure C in terms of the res of Figure A nd Figure B. 3. The figure denotes trpezium of re 33 cm 2 tht hs been formed by joining two tringles together. Find the re of the tringle which is shded. 4. The figure denotes prllelogrm of re 20 cm 2. Its perimeter is 64 cm. Determine the following bsed on the informtion tht is given. (i) The length of the side CD. (ii) The length of the side BC. 66 D A C B
6. Are of sector of circle We considered how the perimeter of sector of circle is found in the lesson on Perimeters. Now let us consider how the re of sector of circle is found. r The following tble shows how the re of sector is found when the ngle t the centre of the sector tkes certin specil vlues. Sector Shded Sector s frction of the circle Are of the Sector πr 2 πr 2 4 4 πr2 3 4 3 4 πr2 20 x 20 x 20 x 3 3 πr2 0 x 0 360 0 360 πr2 x 360 360 πr2 67
r x According to the pttern in the tble, the re of the sector of rdius r nd ngle t the centre x is 360 «πr2 Let us consider (through the following exmples) how the re of sector is found using this result. In the exmples nd exercises of this chpter, the vlue of π is tken s 22 7. Exmple Find the re of the sector in the following figure. Are 45 x Are is 7. Exmple 2 2 If the re of the sector in the figure is, find the rdius of the corresponding circle. 9 Let us tke the rdius s r cm. Are r 40 x Rdius is. 68
Exercise 6.. Find the re of ech sector. (i) (ii) (iii) (iv) 60 x 2 cm 3 cm 280 x 2. The res of the two sectors of circles given below re 7 2 nd 462 cm 2 respectively. Find the ngle t the centre of ech sector. (i) 0'5 cm (ii) 2 cm 3. The res of the two sectors of circles given below re 792 cm 2 nd respectively. For ech sector, find the rdius of the corresponding circle. (i) (ii) 280 x 80 x 6.2 Plne figures contining sectors of circles Let us consider the res of plne figures formed by sectors of circles nd other simple plne figures such s rectngles nd tringles being joined together. Exmple The following denotes plne figure consisting of squre nd semi-circle. Find its re. 69
Are of the squre = x = 9 2 Since the dimeter of the semi-circle is equl to the length of side of the squre, The rdius of the circle is 4 2 =. Are of the semi-circle Are of the compound plne figure Exmple 2 The figure denotes compound plne figure consisting of rectngle nd two sectors of circle. Find its re. 0 cm Are of the rectngle Are of sector πr Are of both sectors Are of the compound plne figure 70
Exmple 3 The shded portion is obtined by cutting 4 of circle from rectngulr lmin. Find the re of the shded portion using the given dt. 20 cm Are of the rectngle = 20 7 = 40 cm 2 Are of the sector = ''' = 38.5 cm 2 Are of the shded portion = 40 38.5 = 0.5 cm 2 Exercise 6.2. The following is compound plne figure consisting of squre nd two semi-circles. (i) Find the re of the squre. (ii) Find the rdius of semi-circulr portion. (iii) Find the totl re of the two semi-circulr portions. (iv) Find the re of the compound plne figure. 2. The shded portion in the figure ws obtined by cutting out two semi-circulr prts from rectngulr piece of pper. 25 cm (i) Find the re of the rectngle. (ii) Find the totl re of the two semi-circulr prts. (iii) Find the re of the shded portion. 7
3. The following is compound plne figure consisting of prllelogrm nd sector of circle. 30 cm 44 cm (i) Find the re of the prllelogrm. (ii) Find the re of the sector. (iii) Find the re of the compound figure. 4. The figure denotes circulr lmin of rdius 28 cm. The two sectors in the figure re to be cut out. Find the re of the remining portion fter the two sectors hve been cut out. 50 x 5. The following is figure consisting of two sectors of circles. Show tht the rtio of the re of the smller sector to the lrger sector is : 3. 2 cm 5 x 45 x 6. Bsed on the mesurements given in the figure, show tht the re of the unshded region of the lrger sector is seven times the re of the shded sector. Summry The re of sector of circle of rdius r with ngle t the centre x is 360 «πr2 ' 72
Miscellneous Exercise. Find the re of the shded portion in ech of the figures given below which re formed by sectors of circles. (i) (ii) (iii) 60 x 30 60 x x 3'5 cm 2. Find the re of the shded region. The curved line in the figure is the rc of semi-circle. 3. The figure denotes two semi-circles. Show tht the rtio of the un-shded region to the shded region in the figure is 5 : 7. 60 x 60 x 5. A sketch of the re in front of commemortive plque is given in the figure. Grss hs been grown in the three portions within the semi-circle which re in the shpe of sectors of circles, while white snd hs been spred in the other regions. The rdius of ech sector within the semi-circle is 84 cm. 2' m (i)wht is the rdius of the semi-circle in centimetres? (ii) Find the re of the semi-circulr prt in cm 2. (iii) Find the re of one of the sectors with ngle t the centre equl to 30 o. (iv) Find the ngle t the centre of the lrge sector if its re is 848 squre centimetres more thn the sum of the res of the other two sectors. 73