CBSE Class 10 Areas related to Circles Sure Shot Questions Set E

Areas Related to Circles

Topics Covered

• Perimeter and Area of a Circle
• Areas of Sector and Segment of a Circle

1. Perimeter and Area of a Circle

As we know that a circle is a closed curve consisting of a set of all those points of the plane which are at a constant distance from a fixed point in the plane. The fixed point is called its centre. The constant distance is called its radius. It is the line segment joining any point on the boundary (circumference) to centre. The boundary (or perimeter) of a circle is called its circumference.

• Circumference of a circle = \( 2\pi r \) (unit)
• Area of a circle = \( \pi r^2 \) unit\(^2\)
• Area of the circular path formed by two concentric circles of radii \( r_1 \) and \( r_2 \) (\( r_1 > r_2 \)) = \( \pi r_1^2 – \pi r_2^2 = \pi (r_1^2 – r_2^2) \) (unit)\(^2\)
• The distance travelled (covered) by a wheel in 1 round = its circumference = \( 2\pi r \) (unit)
• Total distance covered by a wheel = its circumference \( \times \) number of rounds taken by it.
• Number of rounds made by a wheel = \( \frac{\text{Total distance covered}}{\text{Its circumference}} \)
• Speed of the wheel = \( \frac{\text{Total distance covered}}{\text{Time taken}} \)
• If speed = \( x \) km/hr, then speed = \( x \times \frac{5}{18} \) m/s and if speed = \( x \) m/s, then speed = \( x \times \frac{18}{5} \) km/hr.

Note: Unless stated otherwise, the value of \( \pi \) is to be taken as \( \frac{22}{7} \).

Question. The area of the circle, the circumference of which is equal to the perimeter of a square of side 11 cm is
(a) 122 cm\(^2\)
(b) 144 cm\(^2\)
(c) 154 cm\(^2\)
(d) 180 cm\(^2\)
Answer: (c) 154 cm\(^2\)

Question. The area of a ring shaped region enclosed between two concentric circles of radii 20 cm and 15 cm is
(a) 330 cm\(^2\)
(b) 415 cm\(^2\)
(c) 520 cm\(^2\)
(d) 550 cm\(^2\)
Answer: (d) 550 cm\(^2\)

Question. If the perimeter and area of a circle are numerically equal; its radius will be
(a) 1 unit
(b) 2 units
(c) 4 units
(d) None of the options
Answer: (b) 2 units

Question. A wheel has diameter 84 cm. Number of complete revolutions must it make to cover 792 metres will be
(a) 100
(b) 160
(c) 220
(d) 300
Answer: (d) 300

Question. 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 a speed of 66 km per hour?
(a) 2275
(b) 2650
(c) 3815
(d) 4375
Answer: (d) 4375

Exercise 7.1

Question. 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
Answer: (b) 14 : 11

Question. The area of the square that can be inscribed in a circle of radius 8 cm is
(a) 256 cm\(^2\)
(b) 128 cm\(^2\)
(c) \( 64\sqrt{2} \) cm\(^2\)
(d) 64 cm\(^2\)
Answer: (b) 128 cm\(^2\)

Question. 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) 50 cm

Question. If the area of circle is numerically equal to twice its circumference, then the diameter of the circle is
(a) 4 units
(b) 6 units
(c) 8 units
(d) 12 units
Answer: (c) 8 units

Question. The perimeter (in cm) of a square circumscribing a circle of radius a cm is
(a) \( 2a \)
(b) \( 4a \)
(c) \( 6a \)
(d) \( 8a \)
Answer: (d) \( 8a \)

Question. What is 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?
(a) 20 cm
(b) 30 cm
(c) 50 cm
(d) 80 cm
Answer: (c) 50 cm

Question. What is the area of the circle that can be inscribed in a square of side 6 cm?
(a) \( 9\pi \) cm\(^2\)
(b) \( 11\pi \) cm\(^2\)
(c) \( 16\pi \) cm\(^2\)
(d) \( 15\pi \) cm\(^2\)
Answer: (a) \( 9\pi \) cm\(^2\)

Question. The area of a quadrant of a circle whose circumference is 25 cm is
(a) 24 cm\(^2\)
(b) 28 cm\(^2\)
(c) 32.5 cm\(^2\)
(d) 38.5 cm\(^2\)
Answer: (d) 38.5 cm\(^2\)

Question. The areas of two circles are in the ratio 9 : 4, then what is the ratio of their circumferences?
(a) 1 : 2
(b) 2 : 1
(c) 3 : 2
(d) 2 : 3
Answer: (c) 3 : 2

Question. The cost of fencing a circular field at the rate of ₹ 24 per metre is ₹ 5280. The radius of the field is
(a) 15 m
(b) 35 m
(c) 25 m
(d) 30 m
Answer: (b) 35 m

Question. The radii of two circles are 8 cm and 6 cm respectively. The radius of the circle having area equal to the sum of the areas of the two circles is
(a) 5 cm
(b) 10 cm
(c) 12 cm
(d) 15 cm
Answer: (b) 10 cm

Question. An athlete runs on a circular track of radius 49 m and covers a distance of 3080 m along its boundary. How many rounds has he taken to cover this distance? \( [\text{Take } \pi = \frac{22}{7}] \)
(a) 5
(b) 8
(c) 10
(d) 15
Answer: (c) 10

Question. The area of the largest triangle that can be inscribed in a semi-circle of radius r units will be
(a) \( r \) sq. units
(b) \( \frac{r}{2} \) sq. units
(c) \( r^2 \) sq. units
(d) \( 2r \) sq. units
Answer: (c) \( r^2 \) sq. units

Question. The area of circle whose circumference is 22 cm is
(a) \( \frac{32}{2} \) cm\(^2\)
(b) \( \frac{45}{2} \) cm\(^2\)
(c) \( \frac{55}{2} \) cm\(^2\)
(d) \( \frac{77}{2} \) cm\(^2\)
Answer: (d) \( \frac{77}{2} \) cm\(^2\)

Question. A road which is 7 m wide surrounds a circular park whose circumference is 88 m. The area of the road is
(a) 220 m\(^2\)
(b) 340 m\(^2\)
(c) 550 m\(^2\)
(d) 770 m\(^2\)
Answer: (d) 770 m\(^2\)

B. Assertion-Reason Type Questions

In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Choose the correct choice as:
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.

Question. Assertion (A): If the circumference of a circle is 176 cm, then its radius is 28 cm.
Reason (R): Circumference = \( 2\pi \times \text{radius} \).
Answer: (a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

Question. Assertion (A): If the outer and inner diameter of a circular path is 10 m and 6 m respectively, then area of the path is \( 16\pi \) m\(^2\).
Reason (R): If \( R \) and \( r \) be the radius of outer and inner circular path respectively, then area of circular path = \( \pi(R^2 – r^2) \).
Answer: (a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

2. Areas of Sector and Segment of a Circle

A sector is a part of the circular region which is enclosed by two radii and the corresponding arc. Hence, OABC is a minor sector and OCDA is a major sector. \( \angle AOC \) is called the angle of sector.

• Area of the sector of a circle of radius \( r \) with central angle \( \theta = \frac{\theta}{360^\circ} \times \pi r^2 \), where \( \theta \) is measured in degrees.
  OR
  Area of the sector = \( \frac{1}{2} \times \text{length of arc} \times \text{radius} = \frac{1}{2} lr \)
• Length of the arc of the sector of a circle of radius \( r \) with central angle \( \theta = \frac{\theta}{360^\circ} \times 2\pi r \), where \( \theta \) is measured in degrees.
• Area of the minor segment APB of the circle in the given figure = area of sector OAPB – area of \( \triangle OAB = \frac{\theta}{360^\circ} \times \pi r^2 – \frac{1}{2} r^2 \times \sin \theta \)
• Area of the major sector of a circle of radius \( r = \pi r^2 \) – area of the corresponding minor sector.

Question. The area of a sector of a circle with radius 6 cm if angle of the sector is 60° is
(a) \( 15\frac{2}{3} \) cm\(^2\)
(b) \( 16\frac{1}{2} \) cm\(^2\)
(c) \( 18\frac{6}{7} \) cm\(^2\)
(d) \( 19\frac{3}{8} \) cm\(^2\)
Answer: (c) \( 18\frac{6}{7} \) cm\(^2\)

Question. A chord AB of a circle of radius 10 cm subtends a right angle at the centre. The area of the minor-sector is \( [\text{Take } \pi = 3.14] \)
(a) 38.5 cm\(^2\)
(b) 42 cm\(^2\)
(c) 78.5 cm\(^2\)
(d) 82 cm\(^2\)
Answer: (c) 78.5 cm\(^2\)

Areas of Combinations of Plane Figures

Question. A square ABCD is inscribed in a circle of radius 10 units. The area of the circle, not included in the square is (Take \( \pi = 3.14 \))
(a) 84 cm\(^2\)
(b) 108 cm\(^2\)
(c) 114 cm\(^2\)
(d) 122 cm\(^2\)
Answer: (c) 114 cm\(^2\)

Question. The area of an equilateral triangle is 17320.5 cm\(^2\). With each vertex as centre, a circle is described with radius equal to half the length of the side of the triangle. The area of the triangle not included in the circles is (Use \( \pi = 3.14 \) and \( \sqrt{3} = 1.73205 \)).
(a) 1620.51 cm\(^2\)
(b) 1810.25 cm\(^2\)
(c) 2430.60 cm\(^2\)
(d) None of the options
Answer: (a) 1620.51 cm\(^2\)

Question. In the given figure, PQ = 24 cm, PR = 7 cm and O is the centre of the circle. The area of the shaded portion is
(a) 132.58 cm\(^2\)
(b) 148.20 cm\(^2\)
(c) 154.36 cm\(^2\)
(d) 161.54 cm\(^2\)
Answer: (d) 161.54 cm\(^2\)

Question. In a circular table cover of radius 32 cm, a design is formed having an equilateral triangle ABC in the middle, as shown below. The area of the design is
(a) 777.36 cm\(^2\)
(b) 1888.11 cm\(^2\)
(c) 2010.54 cm\(^2\)
(d) None of the options
Answer: (b) 1888.11 cm\(^2\)

Question. The area of the shaded portion in the figure, given below, where a circular arc of radius 6 cm has been drawn with vertex O of an equilateral triangle OAB of side 12 cm as centre is
(a) 156.64 cm\(^2\)
(b) 188.46 cm\(^2\)
(c) 256.64 cm\(^2\)
(d) 310.25 cm\(^2\)
Answer: (a) 156.64 cm\(^2\)

Question. ABCD is a square of side 4 cm. At each corner of the square, a quarter circle of radius 1 cm, and at the centre, a circle of radius 1 cm, are drawn, as shown in the given figure. The area of the shaded region is
(a) 8.46 cm\(^2\)
(b) 7.25 cm\(^2\)
(c) 9.71 cm\(^2\)
(d) 10.43 cm\(^2\)
Answer: (c) 9.71 cm\(^2\)

Question. In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. The length of the arc is
(a) 11 cm
(b) 22 cm
(c) 27 cm
(d) 44 cm
Answer: (b) 22 cm

Question. An arc of length 15.7 cm subtends a right angle at the centre of the circle. Then the radius of the circle is
(a) 20 cm
(b) 10 cm
(c) 15 cm
(d) 12 cm
Answer: (b) 10 cm

Question. The angle described by a minute hand in 5 minutes is
(a) 30°
(b) 60°
(c) 90°
(d) None of the options
Answer: (a) 30°

Question. The area of the sector in the following figure showing a chord AB of a circle of radius 18 cm subtending an angle of 60° at the centre O is [Take \( \pi = 3.14 \)]
(a) 151.31 cm\(^2\)
(b) 169.56 cm\(^2\)
(c) 173.33 cm\(^2\)
(d) None of the options
Answer: (b) 169.56 cm\(^2\)

Question. The given figure is a sector of circle of radius 10.5 cm. The perimeter of the sector is [Take \( \pi = \frac{22}{7} \)]
(a) 32 cm
(b) 44 cm
(c) 54 cm
(d) None of the options
Answer: (a) 32 cm

Question. In a circle of diameter 42 cm,if an arc subtends an angle of 60° at the centre where \( \pi = \frac{22}{7} \), then what will be the length of arc?
(a) 11 cm
(b) 20 cm
(c) 22 cm
(d) 28 cm
Answer: (c) 22 cm

Question. A horse is tied to a pole with 28 m long rope. The perimeter of the field where the horse can graze is (Take \( \pi = 22/7 \))
(a) 60 cm
(b) 85 cm
(c) 124 cm
(d) 176 cm
Answer: (d) 176 cm

Question. A car has two wipers which do not overlap. Each wiper has a blade of length 21 cm sweeping through an angle 120°. The total area cleaned at each sweep of the blades is [Take \( \pi = \frac{22}{7} \)]
(a) 360 cm\(^2\)
(b) 448 cm\(^2\)
(c) 556 cm\(^2\)
(d) 924 cm\(^2\)
Answer: (d) 924 cm\(^2\)

Question. The area of the shaded region in the given figure, if AC = 24 cm, BC = 10 cm and O is the centre of the circle is [Take \( \pi = 3.14 \)]
(a) 128.56 cm\(^2\)
(b) 145.33 cm\(^2\)
(c) 248.16 cm\(^2\)
(d) None of the options
Answer: (b) 145.33 cm\(^2\)

Question. A piece of wire 22 cm long is bent into the form of an arc of a circle subtending an angle of 60° at its centre. The radius of the circle is [Take \( \pi = \frac{22}{7} \)]
(a) 7 cm
(b) 14 cm
(c) 21 cm
(d) 28 cm
Answer: (c) 21 cm

Question. The short and long hands of a clock are 4 cm and 6 cm long respectively. The sum of distances travelled by their tips in 2 days is
(a) 1148 cm
(b) 1426.35 cm
(c) 1910.85 cm
(d) None of the options
Answer: (c) 1910.85 cm

Question. The area of the sector of a circle of radius 5 cm, if the corresponding arc length is 3.5 cm is
(a) 3.25 cm\(^2\)
(b) 8.75 cm\(^2\)
(c) 4.60 cm\(^2\)
(d) 5.50 cm\(^2\)
Answer: (b) 8.75 cm\(^2\)

Question. The area of the largest circle that can be drawn inside the given rectangle of length ‘a’ cm and breadth ‘b’ cm (a > b) is
(a) \( \frac{1}{2}\pi b^2 \) cm\(^2\)
(b) \( \frac{1}{3}\pi b^2 \) cm\(^2\)
(c) \( \frac{1}{4}\pi b^2 \) cm\(^2\)
(d) \( \pi b^2 \) cm\(^2\)
Answer: (c) \( \frac{1}{4}\pi b^2 \) cm\(^2\)

Question. All the vertices of a rhombus lie on a circle. The area of the rhombus, if the area of the circle is 1256 cm\(^2\) is [Use \( \pi = 3.14 \)]
(a) 300 cm\(^2\)
(b) 600 cm\(^2\)
(c) 800 cm\(^2\)
(d) 900 cm\(^2\)
Answer: (c) 800 cm\(^2\)

Question. The difference of the areas of two segments of a circle formed by a chord of radius 5 cm subtending an angle of 90° at the centre is
(a) \( (\frac{25\pi}{4} - \frac{25}{2}) \) cm\(^2\)
(b) \( (\frac{15\pi}{4} - \frac{7}{2}) \) cm\(^2\)
(c) \( (\frac{7\pi}{4} - \frac{3}{2}) \) cm\(^2\)
(d) None of the options
Answer: (a) \( (\frac{25\pi}{4} - \frac{25}{2}) \) cm\(^2\)

B. Assertion-Reason Type Questions

Question. Assertion (A): In a circle of radius 6 cm, the angle of a sector is 60°. Then the area of the sector is \( 18\frac{6}{7} \) cm\(^2\).
Reason (R): Area of the circle with radius r is \( \pi r^2 \).
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: (b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).

Question. Assertion (A): The length of the minute hand of a clock is 7 cm. Then the area swept by the minute hand in 5 minute is \( 12\frac{5}{6} \) cm\(^2\).
Reason (R): The length of an arc of a sector of angle \( \theta \) and radius \( r \) is given by \( l = \frac{\theta}{360^\circ} \times 2\pi r \).
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: (b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).