Exercise 11.1
Multiple Choice Questions (Q. No. 1-5)
Question. 1 If the sum of the areas of two circles with radii R₁ and R₂is equal to the area of a circle of radius R, then

Question. 2 If the sum of the circumferences of two circles with radii R₁and R₂is equal to the circumference of a circle of radius R, then
(a) R₁ + R₂ = R
(b) R₁ + R₂ > R
(c) R₁ + R₂ < R
(d) Nothing definite can be said about the relation among R₁, R₂ and R.
Solution. (a)        
Explanation:
According to the question,
Circumference of circle with radius R= Circumference of circle with radius R₁ + Circumference of circle with radius R₂

Question. 3 If the circumference of a circle and the perimeter of a square are equal, then
(a) Area of the circle = Area of the square
(b) Area of the circle > Area of the square
(c) Area of the circle < Area of the square
(d) Nothing definite can be said about the relation between the areas of the circle and square
Solution. (b)
Explanation:
According to the question,
Circumference of a circle = Perimeter of a square

Question. 4 Area of the largest triangle that can be inscribed in a semi-circle of radius r units is

Solution. (a)
Explanation:
The largest triangle that can be inscribed in a semi-circle is the one formed by joining two end points of diameter AB to a point C on the circumference of that semi-circle.

Question. 5 If the perimeter of a circle is equal to that of a square, then the ratio of their areas is
(a) 22:7 
(b) 14:11              
(c) 7:22 
(d) 11:14
Solution. (b)
Explanation:
According to the question,
Perimeter of a circle = Perimeter of a square

Exercise 11.1
Multiple Choice Questions (Q. No. 6-10)
Question. 6 It is proposed to build a single circular park equal in area to the sum of areas of two circular parks of diameters 16 m and 12 m in a locality. The radius of the new park would be
(a) 10m 
(b) 15m 
(c) 20m 
(d) 24m
Answer. (a)          
Explanation:
According to the question,
Area of bigger circular park, A = Area of first circular park, A₁ + Area of second circular park, A₂

Question. 7 The area of the circle that can be inscribed in a square of side 6 cm is
(a) 36pcm²          
(b) 18p cm²          
(c) 12p cm²          
(d) 9p cm²
Answer. (d)
Explanation:

Question. 8 The area of the square that can be inscribed in a circle of radius 8 cm is

Answer. (b)
Explanation:
Let ABCD be a square inscribed in a circle of radius 8 cm and centre O.

Question. 9 The radius of a circle whose circumference is equal to the sum of the circumferences of the two circlesof diameters 36 cm and 20 cm is
(a) 56 cm              
(b) 42 cm              
(c) 28 cm              
(d) 16 cm
Answer. (a)
Explanation:

Question. 10 The diameter of a circle whose area is equal to the sum of the areas of the two circles of radii 24 cm and 7 cm is
(a) 31 cm              
(b) 25 cm              
(c) 62 cm              
(d) 50 cm
Answer. (d)          
Explanation:
According to the question,
Area of bigger circle = Area of first circle + Area of second circle

Exercise 11.2
Very Short Answer Type Questions:
Write whether True or False and justify your answer.
Question. 1 Is the area of the circle inscribed in a square of side a cm, πa² cm²? Give reasons for your answer.
Answer. False
Explanation:
Let a circle with centre O and radius r  be inscribed in a square ABCD with side a.

Question. 2 Will it be true to say that the perimeter of a square circumscribing a circle of radius a cm is 8a cm? Give reason for your answer.
Answer. True
Explanation:

Question. 3 In figure,a square is inscribed in a circle of diameter d and another square is circumscribing the circle. Is the area of the outer square four times the area of the inner square? Give reason for your answer.

Thus, the area of the outer square is two times the area of the inner square.
Hence, given statement is false.
Question. 4 Is it true to say that area of segment of a circle is less than the area of its corresponding sector? Why?
Answer. False
Explanation:

It is true only in the case of minor segment. But in case of major segment, it is always greater than the area of sector.
Question. 5 Is it true that the distance travelled by a circular wheel of diameter d cm in one revolution is 2πd cm? Why?
Answer. False
Explanation:
Distance travelled by the wheel in one revolution = Circumference of wheel = 2πr = πd    [∵2r = d]
Hence, the given statement is false.
Question. 6 In covering a distance s m, a circular wheel of radius r m makes s/2pr revolution. Is this statement true? Why?
Answer. Yes, this statement is true.
Explanation:

Question. 7 The numerical value of the area of a circle is greater than the numerical value of its circumference. Is this statement true? Why?
Answer. No.
Explanation:
For 0 < r < 2, then numerical value of circumference is greater than numerical value of area of circle
Let the radius, r = 1
∴ Area of circle = pr² = p(1)² = p
And circumference of circle = 2pr = 2p(1) = 2p
Thus, the area of a circle is less than its circumference.
Question. 8 If the length of an arc of a circle of radius r is equal to that of an arc of a circle of radius 2r, then the angle of the corresponding sector of the first circle is double the angle of the corresponding
sector of the other circle. Is this statement false? Why?
Answer. No, this statement is true.
Explanation:
Let C₁ and C₂ be the two circles with radii r and 2r, respectively.
According to the question,

Therefore,angle of the corresponding sector of circle, C₁ is double the angle of the corresponding sector of circle C₂.
Question. 9 The area of two sectors of two different circles with equal corresponding arc lengths are equal. Is this
statement true? Why?
Answer. False
Explanation:
Since, area of a sector is given as,

Hence, the areas of two sectors with equal corresponding arc lengths, would be same in case both the circles have equal radii.
Question. 10 The areas of two sectors of two different circles are equal. Is it necessary that their
corresponding arc lengths are equal? Why?
Answer. False
Explanation:
As the area of a sector is given as,

Hence, the areas of two sectors of two different circles would be equal only in case both the circles have equal radii and equal corresponding arc lengths.
Question. 11 Is the area of the largest circle that can be drawn inside a rectangle of length a cm and
breadth b cm (a> b)is pb²cm? Why?
Answer. False
Explanation:
Let a circle with centre O be the largest possible circle drawn inside a rectangle ABCD of length a cm and breadth b cm

Question. 12 Circumference of two circles are equal. Is it necessary that their areas be equal? Why?
Answer. True
Explanation:
Given, circumference of two circles are equal.

Therefore, two circles with equal radii will also have equal areas.
Hence, if circumference of two circles are equal, then their corresponding radii will also be equal. So, their areas will also be equal.
Question. 13 Areas of two circles are equal. Is it necessary that their circumferences are equal? Why?
Answer. True
Explanation:
Given, areas of two circles are equal.

Therefore, two circles with equal radii will have equal circumferences.
Hence, if areas of two circles are equal, then their corresponding radii are equal. So, their circumference will also be equal.
Question. 14 Is it true to say that area of a square inscribed in a circle of diameter p cm is p² cm²? Why?
Answer. False
Explanation:
Let ABCD be a square inscribed in a circle of diameter p cm. 

Exercise 11.3
Short Answer Type Questions (Q. No. 1-8)
Question. 1 Find the radius of a circle whose circumference is equal to the sum of the circumference of two circles
of radii 15cm and 18cm.
Solution.
According to the question,
Circumference of bigger circle = Circumference of first circle + Circumference of second circle

Hence, required radius of circle is 33cm.
Question. 2 In figure, a square of diagonal 8 cm is inscribed in a circle. Find the area of the shaded region.

Solution. To find the area of shaded region first we have to find the area of the circle and the inscribed square.
and the radius of circle be ‘r’.

Let the side of the square be a cm.
Given that, length of diagonal of square, AC = 8cm
Thus, radius of circle, r = 8/2 = 4 cm
Here ∠ABC =  90^o
By applying Pythagoras theorem in right angled ΔABC, we get:

Thus, area of shaded region = 18.28 cm
Question. 3 Find the area of a sector of a circle of radius 28cm and central angle 45°.
Solution.
Given, radius of a circle, r = 28cm
And central angle, θ = 45°

Question. 4 The wheel of a motor cycle is of radius 35cm. How many revolutions per minute must the wheel make,
so as to keep a speed of 66 km/h?
Solution.

Question. 5 A cow is tied with a rope of length 14m at the corner of a rectangular field of dimensions 20m ´ 16m.
Find the area of the field in which the cow can graze.
Solution.
The area covered by cow for grazing is in the form of a sector with radius 14m (= length of the rope).

Question. 6 Find the area of the flower bed (with semi-circular ends) shown in figure.

Solution.
The flower bed is a combination of a rectangle and two semicircles as shown in the figure given below:

Question. 7 In figure, AB is a diameter of the circle, AC = 6cm and BC = 8cm. Find the area of the shaded region, (use π = 3.14)

Solution.
Given figure is a combination of a circle and a ΔACB. 
Since, angle in a semi circle is equal to 90°.
∴          ∠C = 90°                                  
Thus, applying Pythagoras theorem in right angled ΔACB,

Solution. Given shaded region is a combination of a rectangle and a semi circle as shown below:

Exercise 11.3
Short Answer Type Questions (Q. No. 9-16)
Question. 9 Find the area of the shaded region in figure.

Solution.
Given diagrams can be represented as a combination of two triangles (inner and outer) and two semi-circles as shown below:                                                                                                                                          


Question. 10 Find the area of the minor segment of a circle of radius 14cm, when the angle of the corresponding sector is 60°.


Hence, the required area of the minor segment is 17.796 cm².
Question. 11 Find the area of the shaded region in figure, where arcs drawn with centres A, B, C and D intersect in pairs at mid-point P, Q, R and S of the sides AB, BC, CD and DA, respectively of a square ABCD. (use π = 3.14)

Solution.
Given diagram is a combination of four quadrants and a square of side 12 cm.
As, P is a mid-poinl of AB.

Question. 12 In figure arcs are drawn by taking vertices A, B and C of an equilateral triangle of side 10 cm, To intersect the sides BC, CA and AB at their respective mid-points D,E and F. Find the area of the shaded region. (use π= 3.14)

Solution.              
Given diagram is a combination of three sectors and an equilateral triangle of side 10 cm.
In equilateral triangle ABC,

Question.13 In figure, arcs have been drawn with radii 14 cm each and with centres P, Q and R. Find the area of the shaded region.

Solution.
Given, radii of each arc, r = 14cm
Let θ₁, θ₂ and θ₃ be the three angles of ΔPQR.

Hence, the required area of the shaded region is 308cm²
Question. 14 A circular park is surrounded by a road 21m wide. If the radius of the park is 105m, then find the area of the road.
Solution.                                                                        
Given, radius of circular park, r = 105 m
Width of road around the park = 21m
Therefore, radius of bigger circle (park + road), R = 105 + 21 = 126m

Question.15 In figure, arcs have been drawn of radius 21cm each with vertices A, B, C and D of quadrilateral ABCD as centres. Find the area of the shaded region.

Hence, required area of the shaded region is 1386cm².
Question. 16 A piece of wire 20cm long is bent into the form of an arc of a circle, subtending an angle of 60° at its centre. Find the radius of the circle.
Solution.
Given that length of arc of circle made from wire, l = 20cm
And central angle, θ = 60°

Exercise 11.4
Long Answer Type Questions (Q. No. 1-7)
Question. 1 The area of a circular playground is 22176 m². Find the cost of fencing this ground at the rate of Rs 50 per m.
Solution.
Given, area of a circular playground = 22176m²

Question. 2 The diameters of front and rear wheels of a tractor are 80cm and 200cm, respectively. Find the number of revolutions that rear wheel will make in covering a distance in which the front wheel makes 1400 revolutions.
Solution.

Question. 3 Sides of a triangular field are 15m, 16m and 17m. With the three corners of the field a cow, a buffalo and a horse are tied separately with ropes of length 7m each to graze in the field. Find the area of the field which
cannot be grazed by the three animals.
Solution.
Here, the area grazed by each animal at three corners of triangular field is in the form of a sector of a circle with radius 7m.


Question. 4 Find the area of the segment of a circle of radius 12cm whose corresponding sector has a centrel angle of 60°. (user π = 3.14)
Solution.
Given, a circle with radius 12 cm and a segment AB with its corresponding sector subtending a centrel angle of 60°.


Question. 5 A circular pond is 17.5m is of diameter. It is surrounded by a 2m wide path. Find the cost of constructing the path at the rate of Rs 25 per m²?
Solution.                                                                        
Given, a circular pond of 17.5 m diameter is surrounded with a 2m wide path.


Question. 6 In following figure, ABCD is a trapezium with AB || DC. AB = 18cm, DC = 32cm and distance between AB and DC = 14cm. If arcs of equal radii 7 cm with centres A, B, C and D have been drawn, then find the area of the shaded region of the figure.


Question. 7 Three circles each of radius 3.5cm are drawn in such a way that each of them touches the other two. Find the area enclosed between these-circles.
Solution.
Given condition can be represented in the form of diagram as given below:


Thus, area of shaded region enclosed between three circles = Area of ΔABC –Total area of three sectors
                                                                                  = 21.217 – 19.25= 1.967cm2
Exercise 11.4
Long Answer Type Questions (Q. No. 8-14)
Question. 8 Find the area of the sector of a circle of radius 5cm, if the corresponding arc length is 3.5cm.
Solution.
Given a sector of a circle of radius (r) 5cm and its corresponding arc length (l) being 3.5 cm.

Question. 9 Four circular cardboard pieces of radii 7cm are placed on a paper in such a way that each piece touches other two pieces. Find the area of the portion enclosed between these pieces.
Solution.
Given, four circular cardboard pieces of radii (r) 7cm are joined together as shown below:

Also, AP and DP are radii at contact point P. So, AD will be perpendicular to the tangent through P. It implies that interior angles of quadrilateral are 90° each.

Also, area of square ABCD = (Side)² = (14)² = 196cm²                           
Thus, area of shaded region = Area of square ABCD –Area of four sectors = 196 – 154 = 42cm²
Question. 10 On a square cardboard sheet of area 784cm², four congruent circular plates of maximum size are placed such that each circular plate touches the other two plates and each side of the square sheet is tangent to two circular
plates. Find the area of the square sheet not covered by the circular plates.
Solution.
Let each side of square be a cm long and radius of each circular plate be r cm.

Question. 11 Floor of a room is of dimensions 5m ´ 4m and it is covered with circular tiles of diameters 50cm each as shown in figure. Find area of floor that remains uncovered with tiles, (use π = 3.14)

Question. 12 All the vertices of a rhombus lie on a circle. Find the area of the rhombus, if area of the circle is 1256
cm², (use π = 3.14)
Solution.
Let A, B, C and D be the vertices of rhombus lying on a circle.
Let the radius of the circle be r.

Question. 13 An archery target has three regions formed by three concentric circles as shown in figure. If the diameters of the concentric circles are in the ratio 1:2:3, then find the ratio of the areas of three regions.

Question. 14 The length of the minute hand of a clock is 5cm. Find the area swept by the minute hand during the
time period 6:05am and 6:40am.
Solution.
Area swept by the minute hand while moving from 6:05am to 6:40am is in the form of a sector of acircle with radius 5 cm.
i.e., Rdius, r = 5 cm
And, time swept by min hand = (6 : 40 am – 6 : 05 am) = 35 min.
Now, angle revolved by the minute hand in 60 minutes = 360°

Exercise 11.4
Long Answer Type Questions (Q. No. 15-20):
Question. 15 Area of a sector of central angle 200° of a circle is 770cm². Find the length of the corresponding arc of this sector.
Solution.
Let r be the radius of the circle.
Given, central angle of sector, q = 200°
And, area of sector = 770 cm²

Question. 16 The central angles of two sectors of circles of radii 7cm and 21cm are respectively 120° and 40°. Find the areas of the two sectors as well as the lengths of the corresponding arcs. What do you observe?
Solution.
Given, radius of first sector, r₁= 7cm
And central angle of first sector, θ₁ = 120°
Also, radius of second sector, r₂ = 21cm
And central angle of second sector, θ₂ = 40°


Hence, we observe that arc lengths of two sectors of two given circles are equal but area of second circle is three times that of the first circle.
Question. 17Find the area of the shaded region given in figure.

Solution.
Given figure is a combination of 4 semi circles, JHM, MGL, LFK and KEJ and two squares JMLK and ABCD.


Question. 18 Find the number of revolutions made by a circular wheel of area 1.54m² in rolling a distance of 176m.
Solution.

Hence, the required number of revolutions made by the circular wheel is 40.
Question. 19 Find the difference of the areas of two segments of a circle formed by a chord of length 5cm subtending an angle of 90° at the centre.
Solution.
Chord AB = 5 cm subtending an angle of 90° at the centre, divides the circle in two segments minor segment ACB and major segment ADB.


Question. 20 Find the difference of the areas of a sector of angle 120° and its corresponding major sector of a circle of radius 21 cm.
Solution.
Solution.
Given, radius of the circle, r = 21cm

Hence, the required difference in areas of two sectors of given circle is 462 cm².