Question Bank — Areas Related to Circles

QUESTION BANK

Areas Related to Circles (NCERT Class 10)

Section A — MCQs (1 mark each)

1. The area of a sector of a circle with radius \(r\) and central angle \(\theta^\circ\) is given by
   (A) \(\frac{\theta}{360} \times 2\pi r\)
   (B) \(\frac{\theta}{360} \times \pi r^2\)
   (C) \(\frac{\theta}{180} \times \pi r^2\)
   (D) \(\pi r^2 - \frac{\theta}{360} \times \pi r^2\)

2. If the area of a sector is \(\frac{1}{4}\) of the area of the circle, the angle of the sector is
   (A) \(45^\circ\) (B) \(60^\circ\) (C) \(90^\circ\) (D) \(120^\circ\)

3. The length of an arc of a sector with radius \(r\) and angle \(\theta^\circ\) is
   (A) \(\frac{\theta}{360} \times \pi r^2\) (B) \(\frac{\theta}{360} \times 2\pi r\) (C) \(\frac{\theta}{180} \times \pi r\) (D) \(2\pi r\)

4. Area of the major sector =
   (A) Area of minor sector (B) \(\pi r^2\) – Area of minor sector (C) Area of segment (D) Area of triangle

5. A chord divides the circle into two segments. The segment corresponding to the smaller arc is called the
   (A) major segment (B) minor segment (C) sector (D) quadrant

6. In the formula for area of segment, we subtract the area of
   (A) the sector (B) the triangle formed by two radii and the chord (C) the major segment (D) the arc

7. If \(\theta = 360^\circ\), the area of the sector equals
   (A) \(\pi r\) (B) \(2\pi r\) (C) \(\pi r^2\) (D) zero

8. The area of the segment is always
   (A) equal to the area of the sector (B) less than the area of the sector (C) greater than the area of the sector (D) equal to \(\pi r^2\)

Assertion-Reason Questions

9. Assertion (A): The area of a sector with central angle \(\theta^\circ\) is \(\frac{\theta}{360}\pi r^2\).
   Reason (R): The area is obtained by the unitary method taking the full circle (\(360^\circ\)) as having area \(\pi r^2\).
   (a) Both A and R are true and R is the correct explanation of A.
   (b) Both A and R are true but R is not the correct explanation of A.
   (c) A is true but R is false.
   (d) A is false but R is true.

10. Assertion (A): Area of major segment = \(\pi r^2\) – Area of minor segment.
    Reason (R): The sum of areas of major and minor segments equals the area of the circle.
    (a) Both A and R are true and R is the correct explanation of A.
    (b) Both A and R are true but R is not the correct explanation of A.
    (c) A is true but R is false.
    (d) A is false but R is true.

Section B — Very Short Answer (2 marks each)

1. Find the length of an arc of a circle of radius 7 cm that subtends an angle of \(60^\circ\) at the centre.

2. A sector of a circle has radius 14 cm and angle \(45^\circ\). Find its area (use \(\pi = 22/7\)).

3. The area of a sector is \(77\) cm\(^2\) and radius is 7 cm. Find the angle of the sector.

4. Write the formula for the area of the segment of a circle in terms of sector and triangle.

5. If the radius of a circle is doubled, how does the area of a sector with fixed angle change?

6. Find the area of the major sector when the minor sector of angle \(30^\circ\) has area \(11\) cm\(^2\).

Section C — Short Answer (3 marks each)

1. Find the area of the minor segment of a circle of radius 10 cm if the chord subtends a right angle at the centre (use \(\pi = 3.14\)).

2. An arc of length 22 cm subtends an angle of \(60^\circ\) at the centre. Find the radius of the circle.

3. A chord of length \(2 \times 7\sqrt{3}\) cm subtends \(60^\circ\) at the centre of a circle of radius 14 cm. Find the area of the corresponding minor segment (use \(\pi = 22/7\)).

4. The minute hand of a clock is 10.5 cm long. Find the area swept by it in 10 minutes.

5. Find the area of the major segment of a circle of radius 21 cm when the central angle is \(120^\circ\) (use \(\pi = 22/7\), \(\sqrt{3} = 1.73\)).

Section D — Long Answer (5 marks each)

1. A horse is tied to a corner of a rectangular field of length 20 m and breadth 15 m with a rope of length 7 m. Find the area the horse can graze inside the field. If the rope length is increased to 14 m, find the increase in grazing area (use \(\pi = 22/7\)).

2. Derive the formula for the area of a sector of a circle with radius \(r\) and central angle \(\theta^\circ\) using the unitary method. Hence find the area of the sector and the corresponding segment when \(r = 21\) cm and \(\theta = 120^\circ\) (use \(\pi = 22/7\)).

3. A circular park of radius 28 m has two straight paths from the centre to the circumference forming an angle of \(90^\circ\). Find the area of the remaining part of the park excluding the sector between the paths. Also find the length of the boundary arc of the excluded sector.

Section E — Case/Source-Based (4 marks each)

Case 1: Clock Design

A circular clock of radius 14 cm has its minute hand sweeping through an angle of \(180^\circ\) in 30 minutes.

(a) Find the length of the arc swept by the minute hand in 30 minutes.
(b) Calculate the area swept by the minute hand in 30 minutes.
(c) If the hour hand also sweeps \(30^\circ\) in the same period, find the total area swept by both hands (they do not overlap).
(d) Find the area of the remaining part of the clock face not swept by either hand.

Case 2: Garden Sprinkler

A garden sprinkler is fixed at one corner of a square lawn of side 20 m. It waters a sector of radius 7 m with central angle \(90^\circ\).

(a) Find the area watered by the sprinkler.
(b) Find the length of the arc that forms the outer boundary of the watered region.
(c) If the angle is increased to \(180^\circ\), find the new watered area.
(d) Calculate the area of the lawn that remains dry when the angle is \(90^\circ\).