AREAS RELATED TO CIRCLES

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1 HPTER 1 Points to Remember : RES RELTE T IRLES 1. circle is a collection of points which moves in a plane in such a way that its distance from a fixed point always remains the same. The fixed point is called the centre and the fixed distance is known as radius of the circle.. rea and circumference of a circle : If r is the radius of the circle, then (i) circumference = r = d where d = r (diameter) (ii) rea = r d 4 1 (iii) rea of a semicircle r 1 (iv) rea of a quadrant 4 r 3. rea of a circular ring : If R and r (R > r) are radii of two concentric circles, then area enclosed by the two circles. = (R r ) 4. Number of revolutions completed by a rotating wheel istance moved circumference 5. If a sector of a circle of a radius r contains an angle of. Then, (i) length of the arc of the sector r 360 (ii) Perimeter of the sector r r 360 (iii) rea of the sector r 360 (iv) rea of the segment = rea of the corresponding segment rea of the corresponding triangle. 1 r r sin, or r r sin cos (i) In a clock, a minute hand rotates through an angle of 6 in one minute. 1 (ii) In a clock, an hour hand rotates through an angle of in one minute. 188 RES RELTE T IRLES MTHEMTIS X r r

2 Example 1. Solution. Example. Solution. ILLUSTRTIVE EXMPLES In the given figure, an archery target marked with its five scoring areas from the centre outwards as Gold, Red, lue, lack and White. The diameters of the region representing Gold score is 1 cm and each of the other bands is 10.5 cm wide. Find the area of each of the five scoring regions. here, White lack lue Red Gold r = radius of the region representing Gold score = 10.5 cm. [NERT] r 1 = radius of the region representing Gold and Red scoring areas = 10.5 cm cm = 1 cm = r cm r = radius of the region representing Gold, Red and lue scoring areas = 1 cm cm = 31.5 cm = 3 r cm. r 3 = radius of the region representing Gold, Red, lue and lack scoring areas = 31.5 cm cm = 4 cm = 4r cm. r 4 = radius of the region representing Gold, Red, lue, lack and white scoring areas Now, = 4 cm cm = 5.5 cm = 5r cm. 1 = rea of the region representing Gold scoring area = (10.5) r cm = rea of the region representing Red Scoring area = (r) r = 3 = 3 1 = cm = cm 3 = rea of the region representing lue Scoring area = (3r) (r) = 5 = 5 1 = cm = 13.5 cm 4 = rea of the region representing lack Scoring area = (4r) (3r) = r = 1 = cm = 45.5 cm 5 = rea of the region representing White Scoring area = (5r) (4r) = 9r = 9 1 = cm = cm The wheels of a car are of diameter 80 cm each. How many complete revolutions does each wheel make in 10 minutes when the car is travelling at the speed of 66 km per hour? [NERT] We have, Speed of the car = 66 km/hr istance travelled by the car in 60 minutes = 66 km istance travelled by the car in 10 minutes km 0 = 11 km = cm MTHEMTIS X RES RELTE T IRLES 189

3 lso, diameter of car wheels = 80 cm 80 radius of car wheels cm=40cm ( r say) ircumference of the wheels r 40 cm istance travelled by the car when its wheels take one complete revolution 40 cm Number of revolutions made by the wheels in 10 minutes istance covered by the car in 10 minutes = istance covered by the car when its wheels make one complete revolution = Hence, each wheel makes 435 revolutions in 10 minutes. Example 3. In a circle of radius 1 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. [NERT] Solution. Let be the centre of the circle of radius 1 cm such that an arc P subtends = 60 at the centre. 60 (i) Length of the arc P r 1 cm = cm cm P (ii) rea of sector 60 P r 1 1 cm = 31 cm RES RELTE T IRLES MTHEMTIS X 1 cm (iii) rea of the segment 1 P r r sin r.sin (1). sin 60 cm cm 1 4 = cm ns.

4 Example 4. 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 figure). Find (i) the area of that part of the field in which the horse can graze. (ii) the increase in grazing area if the rope were 10 m long instead of 5m. (Use = 3.14). [NERT] Solution. Here r = 5 m, = 90 (i) The area of the part EF of the field in Example 5. which horse can graze r m 360 = m (ii) Here R = 10 m, = 90 rea of the part GH of the field in which horse can graze 90 R m 8.50 m H F 5m E 10m Increase in area (EGHF) when length of the rope increases from 5 m to 10 m = 8.50 m m = m ns. 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 the given figure. Find : (i) the total length of the silver wire required. (ii) the area of each sector of the brooch. [NERT] 15 m G MTHEMTIS X RES RELTE T IRLES 191

5 35 Solution. Here, r = 35 mm r mm (i) Total length of wire used in making the silver brooch = circumference of the brooch + 5 diameter of the brooch (ii) = (r + 5 r) mm mm (110 15) mm = 85 mm r The area of each sector of the brooch 10 [ The brooch is divided into 10 equal sectors] mm mm = 96.5 mm ns Example 6. round table cover has six equal designs as shown in the given figure. If the radius of the cover is 8 cm, find the cost of making the designs at the rate of Rs per cm (use 3 1. ). Solution. Here, r = 8 cm and = 60 [ It is a regular hexagon] Required area = 6 area of the segment = 6 {area of the sector area of the equilateral } 8 cm [NERT] (8) (8) cm = 464 cm cm = cm ost of making the design = Rs = Rs ns. Example. In a circular table of radius 3 cm, a design is formed leaving an equilateral triangle in the middle as shown in the figure. Find the area of the design (shaded region). Solution. In, we have cos 60 and sin RES RELTE T IRLES MTHEMTIS X 8 cm

6 Example and 3 3 = 16 cm and = 16 3 cm = = 3 3 cm rea of the shaded region = rea of the circle rea of 3 (3) (3 3) cm 4 3 cm cm = cm ns. The given figure depicts a racing track whose left and right ends arc semicircular. The distance between the inner parallel line segments is 60 m and they are each 106 m long. If the track is 10 m wide, find : (i) the distance around the track along its inner edge. (ii) the area of the track. [NERT] S R 30m 10 m 60 m P 106 m Solution. (i) istance around the track along its inner edge (ii) Q 30m = 106 m + Perimeter of two semi-circle of radius 60 m m 30 m 1 m m m = m rea of the track = (area of the rectangle SR) + (area of the semi-circular track) (40 30 ) m (40 30)(4030) m m = ( ) m = (160) m = 430 m ns. MTHEMTIS X RES RELTE T IRLES 193

7 Example 9. The area of an equilateral triangle is cm. With each vertex of the triangle as centre, a circle is drawn with radius equal to half the length of the side of the triangle. Find the Solution. area of the shaded region. [Use = 3.14 and ]. [NERT] 60 We know, area of an equilateral triangle 3 (side) (side) (side) cm cm 40000cm side = 00 cm 00 radius of each circle drawn cm 100 cm For each minor sector : = 60 rea of each sector, inside the triangle r (100) cm 360 = cm rea of the triangle not included in the circles = rea of triangle 3 (area of each sector inside the triangle) = cm cm = cm ns. 194 RES RELTE T IRLES MTHEMTIS X

8 Example 10. alculate the area of the designed region in the given figure common between two quadrants of circles of radius 8 cm each. 8 cm Solution. rea of the designed region 8 cm F 90 8 cm MTHEMTIS X RES RELTE T IRLES E = (rea of quadrant E rea of ) 1 1 (8) 88cm 4 3 cm 8 cm cm = 36.5 cm ns. PRTIE EXERISE Question based on ircumference and rea of a circle : 1. Find the area of the circle whose circumference is cm.. wire is looped in the form of a circle of radius 8 cm. It is rebent into a square form. Find the length of the side of the square. 3. The radius of the circle is 3 m. What is the circumference of another circle, whose area is 49 times that of the first? 4. Two circles touch externally. The sum of their areas is 130 sq. cm and the distance between their centres is 14 cm. Find the radii of the circles. 5. copper wire, when bent in the form of a square, encloses an area of 484 cm. If the same wire is bent in the form of a circle, find the area enclosed by it. [take = /] 6. wire when bent in the form of an equilateral triangle encloses an area of 11 3 cm. If the same wire is bent in the form of a circle, find the area of the circle.. road which is m wide surrounds a circular track whose circumference is 35 m. Find the area of the road. use 8. race track is in the form of a ring whose inner circumference is 35 m and the outer circumference is 396 m. Find the width of the track. 9. The outer circumference of a circular racing track is 0 m. The track is everywhere m wide. alculate the cost of levelling the track at the rate of 50 paisa per sq. m. use 10. The area enclosed between the two concentric circles is 0 cm. If the radius of the outer circle is 1 cm, calculate the radius of the inner circle. 11. Find the radius of a circle whose area is equal to the sum of the areas of the three circles whose radii are 3 cm, 4 cm, and 1 cm.

9 1. garden roller has a circumference of 3 m. How many revolutions does it make in moving 1 m? 13. bicycle wheel makes 5,000 revolutions in moving 11 km. Find the diameter of the wheel. 14. The diameter of the driving wheel of a bus is 140 cm. How many revolutions per minute must the wheel make in order to keep a speed of 66 km/hr? 15. wheel of diameter 4 cm, makes 40 revolutions per minute. Find : (i) the total distance covered by the wheel in one minute. (ii) the speed of the wheel in km/hr. Question based on rea of sector and segment of a circle : 16. Find the area of a quadrant of a circle whose circumference is cm. 1. Find the area of the sector of a circle whose radius is 14 cm and angle of sector is n arc of length 0 cm subtends an angle of 144 at the centre of the circle. Find the radius of the circle. 19. sector of a circle of radius 8 cm contains an angle of 135. Find the area of the sector. 0. The perimeter of a sector of a circle of radius 5. m is. m. Find the area of the sector. 1. The minute hand of a clock is 10 cm long. Find the area of the face of the clock described by the minute hand between 9:00 am and 9:35 am.. In the given figure, there are shown sectors of two concentric circles of radii cm and 3.5 cm. Find the area of the shaded region. use cm cm 3. The given figure shows a quadrant of a circle of radius 10 cm. Find the area of the shaded region and the perimeter of the sector. 10 cm 4. In the given figure, is the centre of the circle of radius 9 cm and = 150. Find : (i) the length of the major arc cm (ii) the area of the major sector. use cm 196 RES RELTE T IRLES MTHEMTIS X

10 5. n arc of a circle of radius 1 cm has a length of 1.6 cm. Find the angle subtended at the centre of the circle. 6. In the given figure, the length of the minor arc is of the circumference of the circle. Find : 4 (i) (ii) If it is given that the circumference of the circle is 13 cm, find the length of the minor arc and the radius of the circle.. Find the area of a segment of a circle of radius cm if the arc of the segment has measure Find the area of the segment of a circle, given that the angle of the sector is 10 and the radius of the circle is 1 cm. Take and chord of a circle of radius 10 cm subtends a right angle at the centre. Find : (i) rea of the minor sector (ii) rea of the minor segment (iii) rea of major sector (iv) rea of major segment use chord of a circle of radius 1 cm subtends an angle of 10 at the centre. Find the area of the corresponding segment of the circle. [use = 3.14, 3 = 1.3] 31. car has two wipers which do not overlap. Each wiper has a blade of length 5 cm sweeping through an angle of 115. Find the total area cleaned at each sweep of the blades. [NERT] 3. 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 = 3.14] [NERT] 33. n umbrella has 8 ribs which are equally spaced (see figure). ssuming umbrella to be a flat circle of radius 45 cm, find the area between the two consecutive ribs of the umbrella. [NERT] MTHEMTIS X RES RELTE T IRLES 19

11 Questions based on combinations of Plane figures : 34. square is inscribed in a circle of radius 10 cm. Find the area of the circle, not included in the square [use = 3.14] 35. In the given figure, is an equilateral triangle inscribed in a circle of radius 4 cm. Find the area of the shaded region. 36. The side of a square is 4 cm. Find the area of (i) the inscribed circle (ii) the circumscribed circle. 3. The following figure shows a rectangle inscribed in a circle. (i) If = 8 cm and = 6 cm, find the area of the circle not included in the rectangle. (ii) If diameter of the circle is 5 cm and = 15 cm, find the area of the circle not included in the given rectangle. 38. The given figure shows a square, inscribed in a quadrant PQ. If = 0 cm, find the area of the shaded region. [NERT] P Q 39. PQRS is a diameter of a circle of radius 6 cm. The lengths PQ, QR and RS are equal, semi-circles are drawn on PQ and QS as diameters as shown in figure. Find the perimeter and area of the shaded region. 198 RES RELTE T IRLES MTHEMTIS X P Q 40. paper is in the form of a rectangle in which = 0 cm and = 14 cm. semi-circular portion with as diameter is cut off. Find the area of the remaining part. R S

12 41. square park has each side of 100 m. t each corner of the park, there is a flower bed in the form of a quadrant of radius 14 m as shown. Find the area of the remaining part of the park. use 14 m 100 m 4. Four equal circles are described about the four corners of a square so that each touches two of the others as shown in figure. Find the area of the shaded region, if each side of the square is 14 cm. 43. P is a quadrant of a circle of radius 14 cm. With as diameter, a semi-circle is drawn as shown in figure. Find the area of the shaded portion. 14 cm 14 cm 44. In the following figure, is an equilateral triangle. ircles are drawn with vertices of the triangle as centres so that every circle touches the remaining two. If the perimeter of the triangle is 84 cm. Find (i) area of sector, inside the triangle, of each circle. (ii) area of the triangle which is not included in the circle. [use = 3.14, 3 = 1.3] MTHEMTIS X RES RELTE T IRLES 199 P 100 m Q

13 45. In the given figure, and are two diameters of a circle (with centre ) perpendicular to each other and is the diameter of the smaller circle. If = cm, find the area of the shaded region. 46. In the given figure, find the area of the shaded region, where is a square of side 10 cm. [use = 3.14]. 00 RES RELTE T IRLES MTHEMTIS X 10 cm 4. Four cows are tethered at four corners of a square plot of side 50 m, so that they just cannot reach one another (see figure). What area will be left ungrazed? 50 m 48. In the given figure, diameter of the biggest semi-circle is 108 cm, and diameter of the smallest circle is 36 cm, calculate the area of the shaded region. 49. Four equal circles are described at the four corners of a square so that each touches two of the others. The shaded area, enclosed between the circles is 50 m 4 cm. Find the radius of the each circle. 50. Three horses are tethered with m long ropes at the three corners of a triangular field having sides 0 m, 34 m and 4 m. Find the area of the plot which can be grazed by the horses. lso, find the area of the plot which remains ungrazed. [SE 001]

14 51. In an equilateral triangle of side 1 cm, a circle is inscribed touching its sides. Find the area of the portion of the triangle not included in the circle. [use and 3.14] 5. In the given figure, is the centre of the bigger circle, and is its diameter. nother circle with as diameter is drawn. If = 54 cm and = 10 cm, find the area of the shaded region. [SE 000] E MTHEMTIS X RES RELTE T IRLES 01 F G 53. In the given figure, is a rectangle with sides 4 cm and 8 cm. Taking 8 cm as the diameter, two semicircles are drawn. Find the area overlapped by the two semi-circles. 8 cm 54. circle has been inscribed in a square of side 4 cm. etermine the left out area. What will be the left out area of the circle if a square is inscribed in the circle? [use = 3.14] 55. Find the area of the shaded region shown in the figure, where a circular arc of radius 6 cm has been drawn with vertex of an equilateral triangle of side 1 cm as centre. 56. In the given figure, is a square. E and F are arcs of a circle whose centre is the centre of the square. If the area of the square is 49 cm, find the area of the shaded region. E F 4 cm

15 5. In the given figure, a piece of cardboard is given in the shape of a trapezium where and = 90. quarter circle FE is removed from the cardboard. Given = = 3.5 cm and E = 1.5 cm. alculate the area of the remaining piece of the cardboard. 1.5 cm F E 3.5 cm 3.5 cm 58. In the given figure, there are three semi-circles,, and having diameter 3 cm each, and another semicircle E having a circle with diameter 4.5 cm. alculate : (i) the area of the shaded region. (ii) the cost of painting the shaded region at the rate of 50 paisa per cm, to the nearest Rs. E 3 cm 4.5 cm 3 cm 59. Find the perimeter and area of the shaded region whose particulars are given below in each of the following : (i) (iii) 14 cm 8 cm 8 cm 8 cm 0 RES RELTE T IRLES MTHEMTIS X (ii) (iv) 3 cm 3.5 cm 14 cm 10 cm 10 cm

16 4 cm (v) (vii) (ix) 1 cm 6 cm 4 cm 8 cm 10 cm 14 cm 14 cm 8 cm cm (vi) (viii) (x) 18 cm 10 cm 33 cm cm 8 cm 10 cm 3.5 cm 60. Find the area of the shaded parts in the following figures, all dimensions being in cm. 6.5 cm 33 cm (i) (ii) cm MTHEMTIS X RES RELTE T IRLES 03

17 (iii) (v) (vii) (ix) given, area of = 5 cm given, = = = cm (iv) (vi) (viii) 14 (x) 14. ircumference of circular track = r Given, r = 353 m r = 56 m rea of road [ (56 ) (56) ]m 1 14 HINTS T SELETE QUESTINS here, radius of each circle = cm [(63) (56) ]m 618 m r 8 cm 04 RES RELTE T IRLES MTHEMTIS X

18 15. istance covered in 1 round r 1cm =13 cm =1.3 m Now, no. of rounds made in 1 minute = 40 The total distance covered by wheel in 1 minute = m = m distance Now, speed of the wheel m/s = 5.8 m/s = km/hr time ngle destriced by the minute hand in 1 minute = 6 ngle described by the minute hand in 35 minutes = (6 35) = 10 rea swept by the minute hand in 35 minutes 10 (10) cm cm Let =, then r r Now, r = 13 r = 1 cm lso, length of minor arc 13 cm = 38.5 cm Since ribs are equally spaced. ngle made by two consecutive ribs at the centre Thus, area between two consecutive ribs = rea of a sector of a circle of radius 45 cm and sector angle cm cm learly. lso, is median of, and is the centroid. : = : 1 4 : = : 1 = cm. Now, In, 4 cm 1 cm 1 cm rea of 3 ( 1) cm 1 3 cm 4 Required rea = rea of circle rea of triangle = (4) cm 1 3 cm = ( ) cm 3. (i) Join to. Here 8 6 cm 10 cm 10 learly, is diameter of the circle. Radius of the circle cm 5 cm Required rea = rea of circle rea of rectangle 39. here, PQ = QR = RS 1 cm 4 cm. 3 QS = QR + RS = 4 cm + 4 cm= 8 cm. (5) cm 8 6 cm (5 48) cm = cm 4 cm MTHEMTIS X RES RELTE T IRLES 05

19 Required perimeter = rc of semi-circle of radius 6 cm + rc of semi-circle of radius 4 cm + rc of semi-circle of radius cm = ( ) cm = 1 cm. lso, Required area = rea of semi-circle with PS as diameter + rea of semi-circle with PQ as diameter rea of semi-circle with QS as diameter 1 [6 4 ] cm 3.1 cm In right, cm 14 cm cm cm Now, Required area = rea PQ = rea Q rea P = rea Q (rea P rea of ) = (rea of semi-circle with as diameter) (rea of a quadrant of circle with ( ) cm (14) cm cm 4 = 98 cm 46. Let us mark the four unshaded regions as I, II, III and IV. Now, rea of I + rea of III = rea of rea of two semi-circles of radius 5 cm each (5) cm 1.5 cm Similarly, rea of II + rea of IV = 1.5 cm So, rea of shaded portion = rea of rea of (I + II + III + IV) 48. learly, Required rea = ( ) cm = 5 cm = rea of semi-circle of radius 54 cm as radius rea of ) 06 RES RELTE T IRLES MTHEMTIS X II I III 10 cm [ rea of semi-circle of radius cm + rea of circle of radius 18 cm] 1 1 (54) cm () (18) cm 1.86 cm 49. learly, rea of square rea of 4 sectors 4 cm 90 4 ( r) 4 r r r r 4 r cm. IV

20 50. rea which can be grazed = sum of areas of 3 sectors r r. r., where r = m r r 1 ( ) 180 r () m m Now, rea of plot = rea of = 336 m [ using heron s formula] rea ungrazed = 336 m m = 59 m. 53. In RMQ, RM 1 cos 60 RQ 4 PRQ = 10 rea overlapped by two semi-circles = (rea of segment PSQ) = (rea of sector RPSQ rea of RPQ) 10 1 (4) PQ RM cm ( RQsin 60 ) cm cm cm learly = 60. rea of shaded region = rea of major sector of circle + rea of (6) cm (1) cm cm 56. learly = 90 here, side of square 49 cm cm. diagonal = cm cm cm. Now, required area = (rea of segment E) 90 1.sin 90 cm 360 m P m R M S 8 cm m Q 4 cm MTHEMTIS X RES RELTE T IRLES 0

21 cm 14 cm 4 5. Required rea = rea of trapezium rea of quadrant 1 1 (3.5 5) 3.5 cm (3.5) cm cm 9.65 cm 5.5 cm 58. Required rea = rea of semi-circle with radius 4.5 cm [rea of circle with diameter 4.5 cm + (rea of semi-circle with diameter 3 cm)] + rea of semi-circle with diameter 3 cm (4.5) cm (1.5) cm (1.5) cm = cm.315 cm cm = 1.35 cm MULTIPLE HIE QUESTINS Mark the correct alternative in each of the following : 1. circular track has an inside circumference of 440 m. If the width of the track is m, the outside circumference is : (a) 441 m (b) 484 m (c) 65 m (d) none of these. The radius of a car wheel is 49 cm. The number of times the wheel of a car rotate in a journey of 195 m is: (a) 55 (b) 600 (c) 65 (d) 5 3. The given figure consists of a rectangle and a semicircle. The area of the figure is : (a) 43 cm (b) 450 cm (c) 53 cm (d) 550 cm 8 cm 4. student takes a rectangular piece of a paper 30 cm long and 1 cm wide. The area of the greatest circle he can cut from the paper is : (a) 46.5 cm (b) cm (c) cm (d) none of these 5. In the given figure, the larger circle has the radius 8 cm with as its centre. The area of the shaded region is : (a) cm (b) 00.5 cm (c) cm (d) none of these 8 cm 08 RES RELTE T IRLES MTHEMTIS X

22 6. The minute hand of a clock is cm long. The area traced out by the minute hand of the clock between 5 : 15 pm to 5 : 35 pm on a day is : 1 (a) 50 cm 3 1 (b) 51 cm 3 1 (c) 5 cm 3 MTHEMTIS X RES RELTE T IRLES 09 (d) none of these. In an equilateral triangle of side 4 cm, a circle is inscribed touching its sides. The area of the remaining portion of the triangle is: (a) cm (b) cm (c) cm (d) cm 8. In the given figure, the area of the shaded segment of the circle with centre is 4 cm 60 (a) 1.45 cm (b).15 cm (c) 3.5 cm (d) 4.5 cm 9. The perimeter of a sector of a circle of radius 9 cm is 33 cm. The area of this sector is : (a) 5.6 cm (b) 61.5 cm (c) 6.5 cm (d) none of these 10. is a quadrant with radius 14 cm and a semi-circle with as diameter. The area of the shaded portion is : (a) 55 cm (b) 66 cm (c) cm (d) 88 cm VERY SHRT NSWER TYPE QUESTINS (1 MRK QUESTINS) 1. If the perimeter of a protractor is cm, what is its radius?. The length of a minute hand of a wall clock is cm. What is the area swept by it in half an hour? 3. What is the area of a sector of a circle with diameter 1 cm and central angle 10? 4. In figure, is a quadrant of a circle of radius 14 cm. What is its perimeter?

23 5. What is the radius of a circle whose circumference and area are numerically equal? 6. In figure, what is the area of shaded region when is a square of side 14 cm.. If an arc forms 10 at the centre of the circle. What is the ratio of its length to the circumference of the circle? 8. The archery target has three concentric circular regions. The diameter of the regions are in the ratio 1 : : 3. What is the ratio of their areas? 9. bicycle wheel makes 10 revolutions in moving 880 cm. What is the radius of the wheel? 10. wire is in the form of a circle of radius 14 cm. It is bent into a square. etermine the side of the square. 11. In figure, given a sector of a circle of radius 10.5 cm. What is the perimeter of the sector? 1. In figure, find the area of the shaded region cm 13. What is the area of the largest triangle that can be inscribed in a semicircle of radius a cm? 14. In figure, is a square of side cm. What is the area of the shaded region? cm 15. In figure, what is the area of the shaded region included between two concentric circles? 14 cm 10.5 cm 10 RES RELTE T IRLES MTHEMTIS X

24 PRTIE TEST M.M : 30 Time : 1 hour General Instructions : Q. 1-4 carry marks, Q. 5-8 carry 3 marks and Q carry 5 marks each. 1. If the area of a semi-circular region is 1,3 cm, find its perimeter.. The length of a minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes. 3. Find the area of the shaded region, if is a square of side 14 cm and P and P are semicircles. P 14 cm 4. wheel has diameter 84 cm. Find how many complete revolutions must it take to cover 1584 meters. 5. horse is placed for grazing inside a rectangular field 80 m by 55 m and is tethered to one corner by a rope 1 m long. n how much area can it graze? 6. The area of an equilateral triangle is 49 3 cm. Taking each angular point as centre, a circle is described with radius equal to half the length of the side of the triangle as shown. Find the area of the triangle not included in the circle.. chord of a circle of radius 6 cm subtends an angle of 60 at the centre. Find the area of the minor segment. 6 cm Find the area of the shaded region, if PQ = 4 cm, PR = cm and is the centre of the circle. Q R P MTHEMTIS X RES RELTE T IRLES 11 6 cm 14 cm

25 9. n athletic track 14 m wide consists of two straight sections 10 m long joining semi-circulars ends whose inner radius is 35 m. alculate the area of the shaded region. 35m 10. Find the area of the shaded portion : 60 cm (i) 10 m 14m 35 cm NSWERS F PRTIE EXERISE cm. 44 cm m cm and 3 cm cm cm. 618 m 8. m 9. Rs cm 1 RES RELTE T IRLES MTHEMTIS X 14m (ii) 8.5 cm cm cm (i) m (ii) km/hr cm 1. cm cm cm m cm cm cm, 35.1 cm 4. (i) 33 cm (ii) cm (i) 105 (ii) 38.5 cm, 1 cm. 14 cm cm 9. (i) 8.5 cm (ii) 8.5 cm (iii) 35.5 cm (iv) 85.5 cm cm cm km cm cm 35. ( ) cm 36. (i) 1386 cm (ii) cm 3. (i) cm (ii) cm cm cm, 3.1 cm cm m 4. 4 cm cm 44. (i) 10.5 cm (ii) 31.3 cm cm cm m cm 3.5 cm

26 49. cm 50. m, 59 m cm 5. 0 cm cm cm ; 4.56 cm cm cm cm 58. (i) 1.35 cm (ii) Rs (i) 88 cm, 154 cm (ii) 44 cm, 4 cm (iii) 16 cm, 13 cm (iv) 34 cm, 61.5 cm (v) 3.6 cm, 9. cm (vi) 31.4 cm, 5.1 cm (vii) 58.9 cm, 5 cm (viii) 189 cm, 169 cm (ix) 50.3 cm, 116 cm (x) 3.8 cm, 4.6 cm 60. (i) 6 cm (ii) 19.3 cm (iii) 46 cm (iv) 38 cm (v) 14.3 cm (vi) 19.5 cm (vii) 31 cm (viii) 336 cm (ix) 31 cm (x) 38 cm NSWERS F MULTIPLE HIE QUESTINS 1. (b). (c) 3. (c) 4. (b) 5. (a) 6. (b). (d) 8. (a) 9. (c) 10. (c) NSWERS F VERY SHRT NSWER TYPE QUESTINS cm. cm cm cm 5. units 6. 4 cm. 1 : : 4 : cm 10. cm cm 1. 4 cm 13. a cm cm cm NSWERS F PRTIE TEST cm cm 3. 4 cm m 6.. cm. 3. cm cm m 10. (i) cm (ii) 31 cm MTHEMTIS X RES RELTE T IRLES 13

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