Lesson-7 AREAS RELATED TO CIRCLES

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1 Lesson- RES RELTE T IRLES Intoduction cicle is a plane figue bounded by one line () such that the distance of this line fom a fixed point within it (point ), emains constant thoughout That is constant. This line is known as the cicumfeence of a cicle (). The fixed point () is the cente of a cicle. The constant distance fom the cente to the cicumfeence is known as the adius of the cicle. ( b adius) () ny line dawn though the cente and teminated both ways by the cicumfeence is called the diamete of a cicle ( diamete) icumfeence and ea icumfeence the peimete of the cicle. ea the space enclosed by the cicle. n-line memoy: Measuement fo icumfeence () Fomula d ea () oelation fomula: P 4 icula Pathway Let be a cicle whose adius. L T * W K 1. Thee is a pathway PLK outside this cicle whose width W ea of cicula pathway W ( + W) (outside) utside Pathway L. Thee is a pathway inside the cicle Width of pathway W ea of cicula pathway W ( W) W P K (inside) Inside Pathway

2 oollay: If R and denote the adii of the oute and inne cicles espectively, which bound a plane cicula ing, it is evident that R W width of the ing ea of ing R (R ) R W (R + ) (R ) W W(R + ) [since W R ] {If we put the value of R( + W), the fomula fo cicula pathway is obtained. In fact aea of ing aea of cicula pathway} hods and cs b a h l a any ac l (say) chod of the ac a height of the ac h E diamete of cicle d half of chod a Length (ac ) half ac 1/ E chod of the half-ac b cental angle by ac c l any pat of the cicumfeence of a cicle. Smalle path is mino ac and lage pat is majo ac. chod of the c a the line segment which joins the two ends of the ac. Height of the c h the length of the pependicula dawn fom the middle point of an ac upon the chod. hod of the Half-c b the length of the chod of the half of an ac. Sectos and segments (a) Secto is a figue bounded by two adii and the ac intecepted between them ea of secto, space enclosed by the secto of a cicle. ental angle, angle contained between the two adii of the secto. Secto aea 360 Length of the c (l) aea of cicle. cicumfeence of cicle 360 Secto l

3 (b) Segment is a figue bounded by a chod and the coesponding ac. Y If the ac is mino, then it is mino segment (X) othewise the segment is a majo segment (Y) X mino segment Y majo segment ea of cicle ea of (mino + majo segments) ental angle X ea of mino segment ea of secto aea of ea of majo segment ea of secto aea of

4 SLVE EXMPLES Ex.1: Find the aea and the cicumfeence of a cicle whose adius is 3.5 m. Using the fomula ea () (3.5) m 38.5 m icumfeence () m 3.5 m Ex.: How many evolutions will a wheel make in tavelling 58 m if its diamete measues 0. m. Using the fomula, icumfeence of wheel d whee d 0. m distance coveed in one evolution No. of evolutions made by wheel No. of evolutions 40 No. of evolution. Hence, no. of evolutions the wheel makes is 40. Total distance tavelled by wheel icumfeence of wheel Ex.3: The diamete of diving wheel of a bus is 1.4 m. How many evolutions pe minute (pm) must the wheel make in ode to keep a speed of 66 km/hou. Using the fomula, No. of evolutions pe minute Total distance tavelled pe minute icumfeence of wheel ( ) Hence, the wheel makes 50 evolutions/minute

5 Ex.4: The hands of a clock ae 10 cm and cm espectively. Find the diffeence between the distance tavesed by thei extemities in 3 days 5 hous. Total time 3 days 5 hous hous. 10 Hou icle Minute icle adius of minute hand M 10 cm adius of hou hand H cm Total distance tavesed by the extemity of hou hand icumfeence No. of evolution H H 1 H cm Total distance tavesed by extemity of minute hand icumfeence No. of evolutions M M M 10 M 4840 cm The equied diffeence M H cm Ex.5: ssuming the cicumfeence of a cicle to be 3 1 times the diamete, find the cicumfeence of the cicle whose aea is 1386 m. Using the coelation fomula whee 1386 m [Since 3 d d ] 13 m Hence, the cicumfeence of the cicle is 13 m Ex.6: cicula gass plot 40 m. in adius is suounded by a ing of gavel. Find the width of the gavel so that the aea of the gass and gavel may be equal?

6 The cicula gass plot has a ing of gavel cutside it. Radius of gass plot 40 m Width of gavel W? It has been given that aea of gass aea of gavel. W ( + W) W + W 0 W + 80 W 1600 [Since 40] W Gavel Gass plot 40 Gavel W W 16.5 m (consideing the +ve value) Hence, the width of the gavel is 16.5 m Ex.: two-wheeled caiage, whose axle tee is 1. m long, is diven aound a cicle; the oute wheel makes 1 1 evolutions fo evey 1 evolutions of inne wheel. The wheels ae each 0.9 m high. Find the cicumfeence of the cicle descibed by the oute wheel. Let and be two wheels - oute and inne espectively. iamete of the wheels height of the wheels 0.9 m (given) istance between two wheels length of a axle tee P path of inne wheel Path of oute wheel X L Y 1. m oute wheel inne wheel YZ 1. m Now, two wheels on tuning ound a path make two cicles - inne (YL) and oute (ZP). If the adius of inne cicle the adius of oute cicle m + 1. m Z Since, the oute wheel makes 1 1 evolution fo evey 1 evolutions of inne wheel, 1 1 icumfeence of oute cicle icumfeence of inne cicle 1 ( 1.).44 3 So, the cicumfeence of the cicle descibed by oute wheel ( + 1.) m

7 Ex.8: man by walking diametically acoss a cicula gass plot, finds that it has taken him 45 sec. less than if he had kept to the path ound the outside. If he walks 90 metes pe minute, find the cicumfeence of the gass plot. K / mete Let KL be the cicula gass plot whose cicumfeence m L (i) point k (i.e., along cicumfeence) tavelling a distance K mete S1 and point (i.e., along diamete) tavelling a distance mete (since cicumfeence diamete) S (say). Walking speed of the peson V 90 mete/min. 90 mete/sec. 60 Using the fomula distance speed time (S 1 S ) V(t 1 t ) [Since t1 t sec.(given)] cm Hence, the cicumfeence is 31.5 Ex.9: The aeas of two concentic cicles ae 154 m and 308 m espectively, find the beadth of the ing. Let the adius of inne cicle m W 154 m. nothe cicle is outside this cicle. If the width of the ing W ea of ing W ( + W) ute cicle Inne cicle W ( + W) W (14 + W) 154 W (14 + W) W.89 m

8 Ex.10: The cicumfeences of two concentic cicles ae 6.83 m and m. Find the aea between the cicles. Let the cicumfeence of oute cicle m cicumfeence of inne cicle m The aea between two cicles 1? Hee, using the coelation fomula fo cicumfeence and aea ( 1 )( 1 4 ) ( ) ( ) m Hence, the aea between the cicles is 01 m. Ex.11: Find the chod of an ac whose height is 4 m, in a cicle of adius 15 m 4 a Hee, the height of an ac h 4 m (known) the diamete of cicle d m the chod of the ac a? Using the fomula, the chod of the ac h( d h) a 4(30 4) a

9 Ex.1: In a cicle of adius 8 m, find the aea of a secto whose angle measues 165. Hee, the adius of cicle 8 m secto angle 165 secto aea? Using the fomula (8) m Hee, the equied aea of a secto is m Ex.13: The adius of a cicle is 5 cm. zone of that cicle has one of its paallel chods coinciding with the diamete and the othe equal to the adius. Find the aea of the zone. Let and be two paallel chods of the cicle. mino sequent Zone Then adius of cicle 5 cm diamete of cicle 150 cm If is cente of cicle, then 5 and is an equilateal, so, using the fomula ea of mino segment 0.09 (cental angle 60 ).09 (5) cm Requied aea between and ea of semicicle ea of mino segment cm Hence, the aea of the equied, zone 836 cm Ex.14: Two cicles touch extenally. The sum of thei aeas is 130 sq. cm. and the distance between thei centes is 14 cm. Find the adii of the cicles. If two cicles touch extenally, then the distance between thei centes is equal to the sum of thei adii. Let the adii of the two cicles be 1 cm and cm espectively. Let 1 and be the centes of the given cicles. Then,

10 [ 1 14 cm (given)] (i) It is given that the sum of the aeas of two cicles is equal to 130p cm (ii) Now, ( 1 + ) (iii) 1 33 Now, ( 1 ) ( 1 ) ( 1 ) 64...(iv) 1 8 Solving (i) and (iv), we get 1 11 cm and 3 cm. Hence, the adii of the two cicles ae 11 cm and 3 cm. Ex.15: Find the aea of the shaded egion in figue whee is a squae of side 10 cm. (use: 3.14) Let us mak the fou unshaded egions as R 1, R, R 3 and R 4. We have, ea of R 1 + ea of R 3 ea of squae ea of two semi-cicles having centes at Q and S cm [ Radius P 5cm] 10cm ( )cm ( ) cm 1.5 cm R Similaly, we have 10cm R 3 R 4 R ea of R + ea of R cm R 1 ea of the shaded egion P 10cm ea of squae (ea of R 1 + ea of R + ea of R 3 + ea of R 4 ) Q 10cm ( ) cm 5 cm

11 1. Find the aea and the cicumfeence of a cicle whose diamete is 14 m.. piece of wie is bent in the shape of an equilateal tiangle of each side 6.6 m. eshaped cicula ing. Find the diamete of the ing. 3. The aea of a cicle is equal to the aea of a squae. ompae thei peimetes. 4. Two men, and, puchase a gindstone 30 cm in diamete fo Rs. 1, of which pays Rs., and pays Rs. 5. Now supposing the inne most 10 cm of the diamete as useless, how many centimetes of the adius may gind down befoe sending the gindstone to? 5. cicula gass plot, whose diamete is 0 m, contains a gavel walk 5 m wide ound it, 15 m fom the edge. Find what it will cost to tuf the gass plot at Rs. pe m. 6. Fou equal cicles ae descibed about the fou cones of a squae so that each touches two of the othes. Find the aea of the space enclosed between the cicumfeences of the cicles; each side of the squae measuing 8 m. SI LEVEL SSIGNMENT 8 S. The height of an ac is 9 cm, and the chod of the ac is 0.3 m. Find the diamete of the cicle. 8. The height of an ac is 1 m and the diamete of the cicle is 84 m. Find the chod of half the ac. b The height of an ac is m and the chod of half the ac is 63 m. Find the distance of the chod of the ac fom the cente of the cicle. b h } x E

12 10. The adius of the cicle is.8 m. Find the length of an ac which subtends an angle of 135 at the cente l 11. The chod of an ac in 19 m and the chod of half the ac is 11 m. Find the length of the ac x 1. The aea of a secto is 80 m, the adius of the cicle is 16 m. Find the angle of the secto E Find the aea of a secto of a cicle, whose adius is 14 cm, and the length of the ac of the secto is 8 cm In a cicle of adius of 1 cm, an ac subtends an angle of 60 at the cente. Find the aea of the mino and majo segment made by this ac K

13 secto of a cicle has aea of 3.85 cm. What is the length of the ac of the secto? 16. If two cicles be descibed on the bounding adii of a quadant of a cicle whose adius is 10 m, as diametes, find the aea of the figue common to both the cicles. E X K Y 1. Find the aea of the shaded egion in figue, if is a squae of side 14 cm and P and P ae semi-cicles. 18. cicle is inscibed in an equilateal tiangle is side 1 cm. touching its sides. Find the adius of the inscibed cicle and the aea of the shaded pat. 19. Fou equal cicles ae descibed about the fou cones of a squae so that each touches two of the othes as shown in figue. cm cm Find the aea of the shaded egion, each side of the squae measuing 14cm. cm cm 0. The diagam shows two acs, and. c is pat of the cicle with cente and adius P. c is pat of the cicle with cente M and adius PM, whee M is the mid-point of PQ. Show that the aea enclosed by the two acs is equal to 5 3 cm

14 1. P is a quadant of a cicle of adis 14 cm. With as diamete, a semicicle is dawn. Find the aea of the shaded potie. Q P 14 cm 14 cm. The minute hand of a clock is 1 cm long. Find the aea descibed by the minute hand on the face of the clock between.0 M and.05 M. 3. If the adjoining figue is a secto of a cicle of adius 10.5 cm. What is the peimete of the secto? [Take /] 60 NSWERS asic Level ssignment m. 6.3 m cm 5. Rs m. 34 cm 8. 4 m m m m m cm, 1346 cm 15.. cm m cm. 5.5 cm 3. 3 cm

MENSURATION-III

MENSURATION-III CIRCLE: A cicle is a geometical figue consisting of all those points in a plane which ae at a given distance fom a fixed point in the same plane. The fixed point is called the cente and