Chapter 11: Areas Related to Circles
CBSE Official PYQs (2015-2026)
Q1. If the perimeter and the area of a circle are numerically equal, then the radius of the circle is :
Q2. If the difference between the circumference and the radius of a circle is 37 cm, then using \(\pi = \frac{22}{7}\), the circumference of the circle (in cm) is :
Q3. If the perimeter of a semi-circular protractor is 36 cm, then its diameter is :
Q4. The area of a sector of angle \(\theta\) (in degrees) of a circle with radius \(R\) is :
Q5. If the area of a circle is numerically equal to twice its circumference, then the diameter of the circle is :
Q6. The ratio of the area of a circle to the area of the square inscribed in it is :
Q7. If the radius of a circle is doubled, then the ratio of the area of the new circle to that of the original circle is :
Q8. The length of the minute hand of a clock is 14 cm. The area swept by the minute hand in 5 minutes is :
Q9. 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 :
Q10. If the sum of the areas of two circles with radii \(R_1\) and \(R_2\) is equal to the area of a circle of radius \(R\), then :
Q11. The area of the largest triangle that can be inscribed in a semi-circle of radius \(r\) is :
Q12. If the circumference of a circle and the perimeter of a square are equal, then :
Q13. The area of a circular path of uniform width \(h\) surrounding a circular region of radius \(r\) is :
Q14. The area of the region between two concentric circles of radii 7 cm and 14 cm is :
Q15. If the diameter of a semi-circular park is 28 m, then its perimeter is :
Q16. The perimeter of a sector of a circle of radius 5.6 cm is 27.2 cm. The angle of the sector is :
Q17. The ratio of the areas of a circle and an equilateral triangle whose diameter and side are equal is :
Q18. A steel wire when bent in the form of a square encloses an area of 121 sq. cm. If the same wire is bent into the form of a circle, then the area of the circle is :
Q19. The area of a sector of a circle of radius 6 cm is \(9\pi \text{ cm}^2\). The length of the corresponding arc of the sector is :
Q20. If the sum of the circumferences of two circles with radii \(R_1\) and \(R_2\) is equal to the circumference of a circle of radius \(R\), then :
Q21. The wheel of a motorcycle is of radius 35 cm. How many revolutions per minute must the wheel make so as to keep a speed of 66 km/h?
Q22. In a circle of radius 21 cm, an arc subtends an angle of \(60^\circ\) at the centre. The length of the arc is :
Q23. \(O\) is the centre of a circle of radius 10 cm. If \(OAPB\) is a quadrant of the circle, its area (take \(\pi = 3.14\)) is :
Q24. If the difference between the area and the circumference of a circle is \(31\) units, then the radius of the circle is :
Q25. If the angle of a sector of a circle of radius 10.5 cm is \(60^\circ\), then its perimeter is :
Q26. If a circle of radius 7 cm is divided into 12 equal sectors, then the area of each sector is :
Q27. The area of a circle whose area and circumference are numerically equal is :
Q28. If the radii of two circles are in the ratio 4 : 5, then the ratio of their areas is :
Q29. The area of a quadrant of a circle of radius \(r\) is :
Q30. Assertion (A): The area of a sector of angle \(60^\circ\) of a circle of radius 6 cm is \(\frac{132}{7} \text{ cm}^2\).
Reason (R): Area of a sector of angle \(\theta\) is given by \(\frac{\theta}{360} \times \pi r^2\).
Q31. Find the area of a sector of a circle of radius 6 cm if angle of the sector is \(60^\circ\). (Use \(\pi = \frac{22}{7}\))
Q32. Find the area of a quadrant of a circle whose circumference is 22 cm. (Use \(\pi = \frac{22}{7}\))
Q33. An arc of a circle is of length \(5\pi\) cm and the sector it bounds has an area of \(20\pi \text{ cm}^2\). Find the radius of the circle.
Q34. Find the area of a sector of a circle of radius 14 cm and central angle \(45^\circ\). (Use \(\pi = \frac{22}{7}\))
Q35. The perimeter of a sector of a circle of radius 14 cm is 46 cm. Find the area of this sector.
Q36. In a circle of radius 10.5 cm, the minor arc is one-fifth of the major arc. Find the area of the major sector.
Q37. Find the area of the minor sector of a circle of radius 14 cm, whose central angle is \(90^\circ\). (Use \(\pi = \frac{22}{7}\))
Q38. The hour hand of a clock is 6 cm long. Find the area swept by it between 11:20 a.m. and 11:55 a.m.
Q39. Find the length of the arc of a circle of radius 14 cm which subtends an angle of \(60^\circ\) at the centre.
Q40. If the perimeter of a semi-circular plot is 108 m, find its area. (Take \(\pi = \frac{22}{7}\))
Q41. In a circle of radius 21 cm, an arc subtends an angle of \(60^\circ\) at the centre. Find the area of the segment formed by the corresponding chord. (Use \(\pi = \frac{22}{7}\) and \(\sqrt{3} = 1.73\))
Q42. A square \(OABC\) is inscribed in a quadrant \(OPBQ\). If \(OA = 20 \text{ cm}\), find the area of the region outside the square but inside the quadrant. (Use \(\pi = 3.14\))
Q43. A chord of a circle of radius 12 cm subtends an angle of \(120^\circ\) at the centre. Find the area of the corresponding segment of the circle. (Use \(\pi = 3.14\) and \(\sqrt{3} = 1.73\))
Q44. \(ABCD\) is a square of side 14 cm. With centres \(A, B, C\) and \(D\), four circles are drawn such that each circle touches externally two of the remaining three circles. Find the area of the region enclosed between these four circles. (Use \(\pi = \frac{22}{7}\))
Q45. A chord of a circle of radius 15 cm subtends an angle of \(60^\circ\) at the centre. Find the areas of the corresponding minor and major segments of the circle. (Use \(\pi = 3.14\) and \(\sqrt{3} = 1.73\))
Q46. Find the area of the region consisting of a circular arc of radius 6 cm drawn with vertex \(O\) of an equilateral triangle \(OAB\) of side 12 cm as centre, and the triangle \(OAB\).
Q47. A car has two wipers which do not overlap. Each wiper has a blade of length 25 cm sweeping through an angle of \(115^\circ\). Find the total area cleaned at each sweep of the blades. (Use \(\pi = \frac{22}{7}\))
Q48. \(AB\) and \(CD\) are two diameters of a circle (with centre \(O\)) perpendicular to each other and \(OD\) is the diameter of the smaller circle. If \(OA = 7 \text{ cm}\), find the area of the region consisting of the smaller circle and the two minor segments bounded by chords \(AC\) and \(BC\). (Use \(\pi = \frac{22}{7}\))
Q49. Case Study 1: Brooch Design Project
A brooch is made with silver wire in the form of a circle with diameter 35 mm. The wire is also used in making 5 diameters which divide the circle into 10 equal sectors.
Based on the above information, answer the following questions :
- Find the total length of the silver wire required. (2 Marks)
- Find the area of each sector of the brooch. (2 Marks)
1. \(285 \text{ mm}\)
2. \(96.25 \text{ mm}^2\)
Q50. Case Study 2: Archery Target Design
An archery target has five concentric scoring regions from the centre outwards as Gold, Red, Blue, Black, and White. The diameter of the central Gold region is 21 cm and each of the other four bands is 10.5 cm wide.
Based on the above information, answer the following questions :
- Find the area of the Gold scoring region. (1 Mark)
- Find the area of the Red scoring region. (2 Marks)
- Find the total area of the scoring target up to the Blue region. (1 Mark)
1. \(346.5 \text{ cm}^2\)
2. \(1039.5 \text{ cm}^2\)
3. \(3118.5 \text{ cm}^2\)
Q51. Case Study 3: Horse Tied to a Peg in a Field
A horse is tied to a peg at one corner of a square shaped grass field of side 15 m by means of a 5 m long rope.
Based on the above information, answer the following questions :
- Find the area of that part of the field in which the horse can graze. (2 Marks)
- Find the increase in the grazing area if the rope were 10 m long instead of 5 m. (Take \(\pi = 3.14\)) (2 Marks)
1. \(19.625 \text{ m}^2\)
2. \(58.875 \text{ m}^2\)
Q52. Long Answer: Area of Shaded Designs
In a circular metal sheet of radius 28 cm, a decorative floral design is carved. The central circular design consists of 6 equal segments bordering the circle. Find the cost of polishing this central circular design at the rate of \(Rs.\;0.35\) per \(\text{cm}^2\). (Use \(\sqrt{3} = 1.73\) and \(\pi = \frac{22}{7}\))