CBSE Class 10 Mathematics Areas related to circles MCQs Set E

Question. The area of a circular ring formed by two concentric circles whose radii are 5.7 cm and 4.3 cm respectively is (Take \(\pi = 3.1416\))
(a) 43.98 sq. cm.
(b) 53.67 sq. cm.
(c) 47.24 sq. cm.
(d) 38.54 sq. cm.
Answer: A
Let the radii of the outer and inner circles be \(r_1\) and \(r_2\) respectively, we have
Area \(= \pi r_1^2 - \pi r_2^2 = \pi(r_1^2 - r_2^2)\)
\(= \pi(r_1 - r_2)(r_1 + r_2)\)
\(= \pi(5.7 - 4.3)(5.7 + 4.3)\)
\(= \pi \times 1.4 \times 10 \text{ sq. cm}\)
\(= 3.1416 \times 14 \text{ sq. cm}\)
\(= 43.98 \text{ Sq. Cm.}\)

Question. A sector is cut from a circular sheet of radius 100 cm, the angle of the sector being \(240^\circ\). If another circle of the area same as the sector is formed, then radius of the new circle is
(a) 79.5 cm
(b) 81.5 cm
(c) 83.4 cm
(d) 88.5 cm
Answer: B
Area of sector \(= 240/360 \times \pi(100)^2 = 20933 \text{ cm}^2\)
Let \(r\) be the radius of the new circle, then
\(20933 = \pi r^2\)
\(r = \sqrt{\frac{20933}{\pi}} = 81.6 \text{ cm}\)

Question. If a circular grass lawn of 35 m in radius has a path 7 m. wide running around it on the outside, then the area of the path is
(a) 1450 m\(^2\)
(b) 1576 m\(^2\)
(c) 1694 m\(^2\)
(d) 3368 m\(^2\)
Answer: C
Radius of outer concentric circle \(= (35 + 7) \text{ m} = 42 \text{ m}\).
Area of path \(= \pi(42^2 - 35^2) \text{ m}^2 = \frac{22}{7}(42^2 - 35^2) \text{ m}^2 = 1694 \text{ m}^2\)

Question. If the area of a semi-circular field is 15400 sq m, then perimeter of the field is:
(a) \(160\sqrt{2}\) m
(b) \(260\sqrt{2}\) m
(c) \(360\sqrt{2}\) m
(d) \(460\sqrt{2}\) m
Answer: C
Let the radius of the field be \(r\).
Then, \(\frac{\pi r^2}{2} = 15400\)
\(\frac{1}{2} \times \frac{22}{7} \times r^2 = 15400\)
\(r^2 = 15400 \times 2 \times \frac{7}{22} = 9800\)
\(r = 70\sqrt{2} \text{ m}\)
Thus, perimeter of the field \(= \pi r + 2r\)
\(= \frac{22}{7} \times 70\sqrt{2} + 2 \times 70 \times \sqrt{2}\)
\(= 220\sqrt{2} + 140\sqrt{2} = 360\sqrt{2} \text{ m}\)

Question. The area of a sector of angle \(p\) (in degrees) of a circle with radius R is
(a) \(\frac{p}{360} \times 2\pi R\)
(b) \(\frac{p}{180} \times \pi R^2\)
(c) \(\frac{p}{720} \times 2\pi R\)
(d) \(\frac{p}{720} \times 2\pi R^2\)
Answer: D

Question. If the sector of a circle of diameter 10 cm subtends an angle of \(144^\circ\) at the centre, then the length of the arc of the sector is
(a) \(2\pi\) cm
(b) \(4\pi\) cm
(c) \(5\pi\) cm
(d) \(6\pi\) cm
Answer: B

Question. The area of the circle that can be inscribed in a square of side 6 cm is:
(a) \(36\pi \text{ cm}^2\)
(b) \(18\pi \text{ cm}^2\)
(c) \(12\pi \text{ cm}^2\)
(d) \(9\pi \text{ cm}^2\)
Answer: D

Question. The sum of the areas of two circles, which touch each other externally, is \(153\pi\). If the sum of their radii is 15, then the ratio of the larger to the smaller radius is
(a) 4:1
(b) 2:1
(c) 3:1
(d) None of these
Answer: A
Let the radii of the two circles be \(r_1\) and \(r_2\), then
\(r_1 + r_2 = 15\) (given) ...(1)
and \(\pi r_1^2 + \pi r_2^2 = 153\pi\) (given) ...(2)
\(r_1^2 + r_2^2 = 153\)
\(r_1^2 + (15 - r_1)^2 = 153\)
On solving, we get \(r_1 = 12, r_2 = 3\)
Required ratio \(= 12:3 = 4:1\)

FILL IN THE BLANK

Question. Length of arc of a sector angle \(45^\circ\) of circle of radius 14cm is ..........
Answer: \(\frac{7}{2}\pi \text{ cm}\)

Question. The boundary of a sector consists of an arc of the circle and the two .........
Answer: radii

Question. .......... is the region between the arc and two radii.
Answer: sector

Question. The region enclosed by an arc and a chord is called the ......... of the circle.
Answer: segment

Question. Perimeter of a semi circle ..........
Answer: \((\pi r + d)\) units

Question. Circumference of a circle is ..........
Answer: \(2\pi r\)

Question. If radius of a circle is 14 cm the area of the circle is ..........
Answer: 616 cm\(^2\)

Question. Area of a circle is ..........
Answer: \(\pi r^2\)

Question. Measure of angle in a semi circle is ..........
Answer: \(90^\circ\)

Question. Length of an arc of a sector of a circle with radius \(r\) and angle with degree measure \(\theta\) is ..........
Answer: \(\frac{\theta}{360} \times 2\pi r\)

TRUE/FALSE

Question. If a sector of a circle of diameter \(21 \text{ cm}\) subtends an angle of \(120^\circ\) at the centre, then its area is \(85.5 \text{ cm}^2\).
Answer: False

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

Question. Area of a segment of a circle is less than the area of its corresponding sector.
Answer: False

Question. A minor sector has an angle '\(\theta\)' subtended at the centre of the circle, whereas major sector has no angle.
Answer: True

Question. Two circles are congruent if their radii are equal.
Answer: True

Question. The perimeter of a circle is generally known as its circumference.
Answer: True

Question. If circumferences of two circles are equal, then their areas are also equal.
Answer: True

Question. Distance moved by a rotating wheel in one revolution is equal to the circumference of the wheel.
Answer: True

Question. The area of the circle inscribed in a square of side \(a \text{ cm}\), is \(\pi a^2 \text{ cm}^2\).
Answer: False

Question. A segment corresponding a major arc of a circle is known as the major segment.
Answer: True

MATCHING QUESTIONS

Question. Two circular flower beds have been shown on two sides of a square lawn ABCD of side \(56 \text{ m}\). If the centre of each circular flowered bed is the point of intersection O of the diagonals of the square lawn, then match the column.
Column-I:
(A) Area of \(\Delta OAB\)
(B) Area of flower bed
(C) Area of sector OAB
(D) Total area
Column-II:
(p) 4032
(q) 784
(r) 448
(s) 1232
Answer: (A) - q, (B) - r, (C) - s, (D) - p.

Question. Match the Column-I formulas with Column-II.
Column-I:
(A) Circumference
(B) Area of a quadrant
(C) Length of the arc of the sector
(D) Perimeter of the sector
(E) Area of the sector
Column-II:
(p) \(2r + \frac{\theta}{360^\circ} \times 2\pi r\)
(q) \(\frac{\theta}{360^\circ} \times \pi r^2\)
(r) \(\frac{\pi r^2}{4}\)
(s) \(\frac{\theta}{360^\circ} \times 2\pi r\)
(t) \(2\pi r\)
Answer: (A) - t, (B) - r, (C) - s, (D) - p, (E) - q.

ASSERTION AND REASON

DIRECTION : In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark 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 : In a circle of radius \(6 \text{ cm}\), the angle of a sector \(60^\circ\). 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 (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
Area of the sector \( = \frac{\theta}{360} \times \pi r^2 = \frac{60}{360} \times \frac{22}{7} \times 6 \times 6 = \frac{132}{7} = 18 \frac{6}{7} \text{ cm}^2\).

Question. Assertion : If the circumference of a circle is \(176 \text{ cm}\), then its radius is \(28 \text{ cm}\).
Reason : Circumference \( = 2\pi \times\) 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
\(C = 2 \times \frac{22}{7} \times r = 176 \Rightarrow r = \frac{176 \times 7}{2 \times 22} = 28 \text{ cm}\)

Question. Assertion : The length of the minute hand of a clock is \(7 \text{cm}\). Then the area swept by the minute hand in \(5\) minutes is \(12 \frac{5}{6} \text{ cm}^2\).
Reason : The length of an arc of a sector of angle \(\theta\) and radius \(r\) is given by \(l = \frac{\theta}{360} \times 2\pi r\)
(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
Area swept in \(5\) minutes \( = \frac{\theta}{360} \times \pi r^2 = \frac{30}{360} \times \frac{22}{7} \times 7 \times 7 = \frac{77}{6} = 12 \frac{5}{6} \text{ cm}^2\). (Angle in \(5\) minutes is \(30^\circ\))

Question. Assertion : A wire is looped in the form of a circle of radius \(28 \text{ cm}\). It is bent into a square. Then the area of the square is \(1936 \text{ cm}^2\).
Reason : Angle described by a minute hand in \(60\) minutes\(= 360^\circ\).
(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
Length of wire \( = 2\pi r = 2 \times \frac{22}{7} \times 28 = 176 \text{ cm}\). Perimeter of square \( = 176 \Rightarrow 4a = 176 \Rightarrow a = 44\). Area \( = (44)^2 = 1936 \text{ cm}^2\).