CBSE Class 10 Areas related to Circles Sure Shot Questions Set 03

MCQ 

Question. The area of a circular ring formed by two concentric circles whose radii are \( 5.7\text{ cm} \) and \( 4.3\text{ cm} \) respectively is (Take \( \pi = 3.14 \))
(a) 44 sq. cm.
(b) 66 sq. cm.
(c) 22 sq. cm.
(d) 33 sq. cm.
Answer: (a) 44 sq. cm.

Question. The diameter of a wheel is \( 1.26\text{ metres} \). How long will it travel in 500 revolutions?
(a) 1492
(b) 2530
(c) 1980
(d) 2880
Answer: (c) 1980

Question. Area of a sector of a circle of radius R, whose central angle is P (in degrees) is given by:
(a) \( \frac{P}{180} \times 2\pi R \)
(b) \( \frac{P}{180} \times \pi R^2 \)
(c) \( \frac{P}{180} \times 2\pi R^2 \)
(d) \( \frac{P}{360} \times 2\pi R \)
Answer: (c) \( \frac{P}{180} \times 2\pi R^2 \)

Question. The area of a circle is 154 cm, then its diameter is
(a) 7cm
(b) 21cm
(c) 40cm
(d) 28cm
Answer: (c) 40cm

Question. The area of the circle that can be inscribed in the square of side 6cm is
(a) \( 18\pi \text{ cm} \)
(b) \( 12\pi \text{ cm} \)
(c) \( 9\pi \text{ cm} \)
(d) \( 14\pi \text{ cm} \)
Answer: (c) \( 9\pi \text{ cm} \)

Question. The perimeter of circular field is \( 242\text{ cm} \). The area of the field is :-
(a) \( 9317\text{ cm}^2 \)
(b) \( 18634\text{ cm}^2 \)
(c) \( 4658.5\text{ cm}^2 \)
(d) None of the options
Answer: (c) \( 4658.5\text{ cm}^2 \)

Question. The difference between the circumference and radius of a circle is \( 37\text{ cm} \). The area of the circle is:-
(a) \( 111\text{cm}^2 \)
(b) \( 184\text{ cm}^2 \)
(c) \( 154\text{ cm}^2 \)
(d) \( 259\text{ cm}^2 \)
Answer: (c) \( 154\text{ cm}^2 \)

Question. The area of the square is the same as the area of the circle. Their perimeter are in the ratio:-
(a) 1:1
(b) \( \pi : 2 \)
(c) \( 2 : \pi \)
(d) None of the options
Answer: (d) None of the options

Question. In making 1000 revolutions, a wheel covers \( 88\text{ Km} \). The diameter of the wheel is:-
(a) \( 14\text{ m} \)
(b) \( 24\text{ m} \)
(c) \( 28\text{ m} \)
(d) \( 40\text{ m} \)
Answer: (c) \( 28\text{ m} \)

Question. The diameter of a wheel is \( 40\text{ cm} \). How many revolutions will it make on covering \( 176\text{ m} \) ?
(a) 140
(b) 150
(c) 160
(d) 166
Answer: (a) 140

Question. A wire is looped in the form of a circle of radius \( 28\text{ cm} \). It is re-bent into a square form. Determine the length of the side of the square:-
(a) \( 42\text{ cm} \)
(b) \( 44\text{ cm} \)
(c) \( 46\text{ cm} \)
(d) \( 48\text{ cm} \)
Answer: (b) \( 44\text{ cm} \)

Question. A road which is \( 7\text{ m} \) wide surrounds a circular park whose circumference is \( 352\text{ m} \). Find the area of the road.
(a) \( 2618\text{ m}^2 \)
(b) \( 2518\text{ m}^2 \)
(c) \( 1618\text{ m}^2 \)
(d) None of the options
Answer: (a) \( 2618\text{ m}^2 \)

Question. A square is inscribed in a circle of radius r. Find the area of the square in \( \text{sq.units} \)
(a) \( 3r^2 \)
(b) \( 2r^2 \)
(c) \( 4r^2 \)
(d) None of the options
Answer: (b) \( 2r^2 \)

Question. Find the area of a right angled triangle, if the radius of its circumscribed circle is \( 2.5\text{ cm} \) and the altitude drawn to the hypotenuse is \( 2\text{ cm} \) long.
(a) \( 5\text{ cm}^2 \)
(b) \( 6\text{ cm}^2 \)
(c) \( 7\text{ cm}^2 \)
(d) None of the options
Answer: (a) \( 5\text{ cm}^2 \)

Question. The perimeter of a sector of a circle of radius \( 5.6\text{ cm} \) is \( 27.2\text{ cm} \). Find the area of the sector.
(a) \( 44\text{ cm}^2 \)
(b) \( 44.6\text{ cm}^2 \)
(c) \( 44.8\text{ cm}^2 \)
(d) None of the options
Answer: (c) \( 44.8\text{ cm}^2 \)

ASSERTION AND REASONING

Question. Assertion : In a circle of radius \( 6\text{ cm} \), the angle of a sector is \( 60^\circ \) Then the area of the sector is \( 18\frac{6}{7}\text{ cm}^2 \).
Reason : 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
(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). The correct formula for area of the sector is \( \frac{\theta}{360} \times \pi r^2 \)

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

Question. Assertion : If a wire of length \( 22\text{ cm} \) is bent in the shape of a circle, then area of the circle so formed is \( 40\text{ cm}^2 \).
Reason : Circumference of the circle = length of the wire.
(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: (d) Assertion (A) is false but reason (R) is true. The assertion is wrong as the area calculated is wrong (Area = \( \frac{77}{2}\text{ cm}^2 \))

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

Question. Assertion : The area of circular playground is \( 22176\text{ m}^2 \) the cost of fencing this ground at the rate Rs. 50 per metre is Rs. \( 26400 \).
Reason : If R and r be the radius of outer and inner circular path, then the area of the ring will be \( \pi(R^2 - r^2) \)
(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).

Short Answer Type -1 

Question. The sum of the areas of two circle, which touch each other externally, is \( 153\pi \). If the sum of their radii is 15, then find the ratio of the larger to the smaller radius .
Answer: 4:1

Question. If the radius of a circle is diminished by 10%, then find how much area will diminished ?
Answer: 36%

Question. The circumference of two circles are in the ratio 2:3. Then find ratio of their areas.
Answer: 4:9

Question. On increasing the diameter of circle by 40%, find how much its area will be increased?
Answer: 96%

Question. The radii of two circles are \( 8\text{ cm} \) and \( 6\text{ cm} \) respectively. Find the radius of the circle having its area equal to the sum of the areas of the two circles.
Answer: \( 10\text{ cm} \)

Question. The minute hand of a clock is \( 12\text{ cm} \) long, find the area swept by minute hand in 35 minutes.
Answer: \( 264\text{ cm}^2 \)

Question. The sum of radii of two circles is \( 140\text{ cm} \) and the difference of their circumferences is \( 88\text{ cm} \). Find the diameters of the circles.
Answer: \( d1 = 154\text{ cm}, d2 = 126\text{ cm} \)

Question. An arc of a circle is of length \( 5\pi\text{ cm} \) and the sector it bounds has an area of \( 25\text{ sq cm} \). Find radius of circle.
Answer: \( 8\text{ cm} \)

Question. Find the area of sector of central angle \( x^\circ \) of a circle with radius \( 4r \).
Answer: \( \frac{2\pi x r^2}{45^\circ} \text{ cm}^2 \)

Question. In a circle of radius \( 21\text{ cm} \), an arc subtends an angle of \( 60^\circ \) at the centre. Find length of the arc.
Answer: \( 22\text{ cm} \)

Short Answer Type 2 

Question. The wheels of a car are of diameter \( 80\text{ cm} \) each. How many complete revolutions does each wheel make in 10 minutes when the car is travelling at a speed of \( 66\text{ km} \) per hour?
Answer: 4375

Question. A field is in the form of a circle. A fence is to be erected around the field. The cost of fencing would be \( Rs2640 \) at the rate of \( Rs12 \) per metre. Then, the field is to be thoroughly ploughed at the cost of \( Rs 0.50 \) per \( \text{m}^2 \). What is the amount required to plough the field?
Answer: Rs. 1925

Question. A car travels \( 1\text{ kilometre} \) distance in which each wheel makes 450 complete revolutions. Find the radius of the its wheels.
Answer: 35.35

Question. The area of enclosed between the concentric circles is \( 770\text{ cm}^2 \). If the radius of the outer circle is \( 21\text{ cm} \), find the radius of the inner circle.
Answer: \( 14\text{ cm} \)

Question. A circular park is surrounded by a rod \( 21\text{ m} \) wide. If the radius of the park is \( 105\text{ m} \), find the area of the road.
Answer: \( 15246\text{ cm}^2 \)

Question. A road which is \( 7\text{ m} \) wide surrounds a circular park whose circumference is \( 352\text{ m} \). Find the area of the road.
Answer: \( 2618\text{ m}^2 \)

Question. Find the radius of a circle whose circumference is equal to the sum of the circumferences of two circles of radii \( 15\text{ cm} \) and \( 18\text{ cm} \).
Answer: \( 33\text{ cm} \)

Question. A pendulum swings through an angle \( 60^\circ \) and describes an arc \( 8.8\text{ cm} \) in length, then find the length of pendulum.
Answer: \( 8.4\text{ cm} \)

Question. Find the area of sector of a circle with radius \( 6\text{ cm} \) and sector angle is \( 60^\circ \).
Answer: \( \frac{132}{7}\text{ cm}^2 \)

Question. The circumference of a circle exceeds the diameter by \( 16.8\text{ cm} \). Find the circumference of the circle.
Answer: \( 24.64\text{ cm} \)

LA TYPE

Question. An archery target has three regions formed by the concentric circles as shown in the 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.
Answer: 1:3:5

Question. The cost of fencing a circular field at the rate of Rs \( 36 \) per \( \text{m} \) is \( 11880 \). The field is to be ploughed at the rate of Rs. \( 0.60 \) per \( \text{m}^2 \). Then find the cost of ploughing the field.
Answer: Rs. 5197.5

CASE STUDY QUESTION 

John had a farm with many animals like cows, dogs, horses etc. He had sufficient grass land for the cows and horses to graze, One day Three of his horses were tied with \( 7\text{metre} \) long ropes at the three corners of a triangular lawn having sides \( 20\text{m}, 34\text{m} \) and \( 42\text{m} \).

Question. (a) Find the area of the triangular lawn .
Answer: \( 336\text{ m}^2 \)

Question. (b) Find the area of the field that can be grazed by the horses.
Answer: \( 77\text{m}^2 \)

Question. (c) The area that cannot be grazed by the horses.
Answer: \( 259\text{m}^2 \)

Raksha Bandhan, is a popular annual rite, or ceremony, which is celebrated in South Asia, and in other parts of the world significantly influenced by Hindu culture. On this day, sisters of all ages tie a talisman, or amulet, called the rakhi, around the wrists of their brothers, symbolically protecting them, receiving a gift in return, and traditionally investing the brothers with a share of the responsibility of their potential care.

Question. Refer to Design A: Rakhi A is made with silver wire in the form of a circle with diameter \( 28\text{mm} \). The wire used for making 4 diameters which divide the circle into 8 equal parts. (a). Find the total length of silver wire required.
Answer: 200 mm

Question. (b) Find the area of each sector of the Rakhi.
Answer: \( 77\text{ mm}^2 \)

Question. Refer to Design B: Rakhi B is made of two colours - Gold and silver. Outer part is made with Gold. The circumference of silver part is \( 44\text{mm} \) and the gold part is \( 3\text{mm} \) wide everywhere. (c) Find the circumference of outer part (golden).
Answer: 62.86 mm