Areas Related to Circles · Class 10 Mathematics: 116 MCQs with answers

Q1. A sector is the region between

Q2. The smaller of the two pieces cut off by two radii is the

Q3. A sector of angle $\theta$ is what fraction of the circle?

Q4. The area of a sector of angle $\theta$ is

Q5. The length of the arc of a sector of angle $\theta$ is

Q6. A quadrant is a sector of angle

Q7. A semicircle is a sector of angle

Q8. If the minor sector has angle 110°, the major sector has angle

Q9. The area of a sector is measured in

Q10. Arc length is measured in

Q11. A circle of radius 7 cm has circumference

Q12. That same circle has area

Q13. If a question gives the DIAMETER, you should

Q14. A windscreen wiper sweeping through an angle traces out

Q15. A sector of angle 360° is

Q16. Define a sector and distinguish the minor sector from the major one.

Q17. Write the formulas for the area of a sector, the length of its arc and its perimeter, and say what each is measured in.

Q18. A segment of a circle is the region between

Q19. A sector is bounded by

Q20. The smaller of the two regions cut off by a chord is the

Q21. The area of a minor segment equals

Q22. The triangle subtracted is formed by

Q23. The syllabus restricts segment problems to central angles of

Q24. The cross-section of liquid in a partly filled horizontal pipe is

Q25. The word 'segment' comes from a Latin verb meaning

Q26. Define a segment of a circle and explain how it differs from a sector.

Q27. The perimeter of a sector equals

Q28. Setting $\theta = 360$ in the sector area formula gives

Q29. A 60° sector of a circle of radius 21 cm has area, with $\pi = \frac{22}{7}$, is

Q30. The arc length of that same sector is

Q31. A quadrant of a circle of radius 14 cm has area, with $\pi = \frac{22}{7}$, is

Q32. The arc of a quadrant of radius $r$ is

Q33. Why do textbook questions so often use a radius of 7, 14 or 21 cm?

Q34. If the radius is 5 cm, the best value of $\pi$ to use is

Q35. The minor and major sectors of a circle have areas that

Q36. Assertion (A): The perimeter of a sector is not the same as the length of its arc. Reason (R): The boundary of a sector includes the two radii as well as the arc.

Q37. The first simplification worth making in a sector calculation is

Q38. An answer of '154 cm' for the area of a sector is

Q39. Which quantity does NOT change when the central angle changes?

Q40. A semicircle of radius 7 cm has area, with $\pi = \frac{22}{7}$, is

Q41. Assertion (A): A sector of 90° has an arc equal to a quarter of the circumference. Reason (R): A circle of radius $r$ has circumference $2\pi r$.

Q42. Find the area and the arc length of a sector of angle 60° in a circle of radius 21 cm. Take $\pi = \frac{22}{7}$.

Q43. Find the perimeter of that same sector, and explain why it differs from the arc length.

Q44. Find the area and the perimeter of a quadrant of a circle of radius 14 cm. Take $\pi = \frac{22}{7}$.

Q45. Explain why textbook questions so often use radii of 7, 14, 21 or 28 cm, and what to do when the radius is not such a number.

Q46. That triangle is always

Q47. The area of triangle $OAB$ with central angle $\theta$ is

Q48. When the central angle is 60°, the triangle is

Q49. When the central angle is 60°, the chord equals

Q50. When the central angle is 90°, the triangle's area is

Q51. In a circle of radius 12 cm a chord subtends 60°. The sector's area, in terms of $\pi$, is

Q52. A segment's area is found by subtracting the triangle from

Q53. The perimeter of a segment is

Q54. The perimeter of a SECTOR, by contrast, is

Q55. A segment answer with no subtraction in it probably means

Q56. A typical segment area has the form

Q57. In a circle of radius 28 cm a chord subtends 90°. The sector's area, with $\pi = \frac{22}{7}$, is

Q58. A segment and the sector it sits in have areas such that the segment is

Q59. In a circle of radius 12 cm, a chord subtending 60° has length

Q60. Assertion (A): At 60° the chord of a circle equals its radius. Reason (R): An isosceles triangle with an apex angle of $60^\circ$ is right-angled.

Q61. Explain how the area of a minor segment is found, and name the triangle involved.

Q62. State the area of triangle $OAB$ for each of the three permitted central angles.

Q63. Across a disc of radius 10 cm a chord makes a right angle where the two radii meet. How large is the lesser of the two pieces? Leave $\pi$ in the answer.

Q64. Why does the syllabus restrict segment problems to central angles of 60°, 90° and 120°?

Q65. The perimeter of that same sector is

Q66. A sector of a circle of radius 6 cm has area $3\pi$ cm². Its angle is

Q67. Doubling the central angle of a sector

Q68. Assertion (A): Doubling the radius of a sector doubles its area. Reason (R): The area of a sector is $\frac{\theta}{360}\pi r^2$.

Q69. Doubling the radius of a sector multiplies its arc length by

Q70. A sector of angle 120° in a circle of radius 21 cm has area

Q71. A sector of angle 45° in a circle of radius 28 cm has arc length

Q72. A sector has arc 11 cm in a circle of radius 21 cm, with $\pi = \frac{22}{7}$. Its angle is

Q73. The perimeter of a quadrant of radius 14 cm is

Q74. Two sectors of the same circle have angles 40° and 80°. Their areas are in the ratio

Q75. Two sectors of the same angle have radii 3 cm and 6 cm. Their areas are in the ratio

Q76. Those same two sectors have arc lengths in the ratio

Q77. Assertion (A): The area of a sector can be found without knowing the radius, provided the angle is given. Reason (R): The area formula contains both $\theta$ and $r$.

Q78. The radian was introduced so that

Q79. A sector of a circle of radius 6 cm has area $3\pi$ cm². Find its central angle.

Q80. A sector of a circle of radius 21 cm has an arc of 11 cm. Find its central angle, taking $\pi = \frac{22}{7}$.

Q81. Explain what happens to a sector's area and arc length when its radius is doubled, keeping the angle fixed.

Q82. The minor sector of a circle of radius 7 cm has angle 90°. Find the area of the major sector, taking $\pi = \frac{22}{7}$.

Q83. (a) Explain why every quantity associated with a sector is $\frac{\theta}{360}$ of the corresponding quantity for the whole circle. (b) Find the area, arc length and perimeter of a sector of angle 60° in a circle of radius 21 cm, taking $\pi = \frac{22}{7}$. (c) Check your formulas by putting $\theta = 360$.

Q84. (a) Explain the difference between the arc of a sector and its perimeter, and why the distinction matters. (b) Find the area and the perimeter of a quadrant of a circle of radius 14 cm, with $\pi = \frac{22}{7}$. (c) A sector of the same circle has arc 11 cm. Find its central angle.

Q85. (a) Describe how to find a sector's central angle when its area is given, and when its arc length is given. (b) A sector of a circle of radius 6 cm has area $3\pi$ cm². Find its angle. (c) Explain why leaving $\pi$ as a symbol made part (b) easier, and when it would be better to substitute a numerical value.

Q86. (a) Explain how a sector's area and arc length respond to a change of angle and to a change of radius. (b) Two sectors of the same circle have angles 40° and 80°; two other sectors of the same angle have radii 3 cm and 6 cm. Give the ratio of areas and of arc lengths in each case. (c) State the units used for each quantity in this chapter and why getting them wrong costs marks.

Q87. When the central angle is 120°, the triangle's area is

Q88. The 60° and 120° triangles of a given circle have areas that are

Q89. A chord of a circle of radius 10 cm subtends 90° at the centre. The minor segment's area is

Q90. In that same problem, the MAJOR segment's area is

Q91. For that 60° sector, the equilateral triangle removed has area

Q92. The major segment's area is best found as

Q93. The length of a chord subtending $\theta$ at the centre is

Q94. Assertion (A): The area of a segment is always a simple fraction of the circle's area. Reason (R): A segment is a sector with a triangle removed, and the triangle is not a fixed fraction of the circle.

Q95. Assertion (A): The perimeter of a segment includes the two radii. Reason (R): The boundary of a segment consists of the arc and the chord.

Q96. A chord subtending 90° in a circle of radius 10 cm has length

Q97. A chord subtending 120° in a circle of radius 6 cm has length

Q98. Why should $\sqrt3$ be kept exact until the last line?

Q99. Inside that 90° sector, the right triangle has area

Q100. So the minor segment there has area

Q101. Using $\frac12 \times \text{base} \times r$ for the triangle's area is

Q102. For a fixed radius, which central angle gives the LARGEST minor segment among the three allowed?

Q103. For a fixed radius, the triangle's area is largest at

Q104. A segment of angle 180° is

Q105. Which quantity is NOT needed to find a minor segment's area?

Q106. Assertion (A): The 60° and 120° triangles of a given circle have the same area. Reason (R): $\sin 120° = \sin 60°$.

Q107. A chord of a circle of radius 20 cm subtends 90°. The minor segment's area, in terms of $\pi$, is

Q108. For that same circle and chord, find the area of the major segment.

Q109. A chord of a circle of radius 21 cm subtends 60° at the centre. Find the area of the minor segment, taking $\pi = \frac{22}{7}$.

Q110. Derive the formula for the length of a chord subtending $\theta$ at the centre, and evaluate it for the three permitted angles.

Q111. Give the perimeter of a segment and explain why it differs from that of a sector.

Q112. Explain why rounding $\sqrt3$ early is dangerous in a segment calculation.

Q113. (a) Distinguish a segment from a sector, and explain how a segment's area is found. (b) A chord of a circle of radius 14 cm subtends a right angle at the centre. Find the areas of the minor and major segments, taking $\pi = \frac{22}{7}$. (c) Find the perimeter of the minor segment, and say why the radii do not appear in it.

Q114. (a) State the area of triangle $OAB$ for each permitted central angle and explain why 60° and 120° give the same answer. (b) A chord of a circle of radius 12 cm subtends 120° at the centre. Find the area of the minor segment, leaving $\pi$ in the answer. (c) Find the length of that chord.

Q115. (a) Explain why a segment's area cannot be written as a fixed fraction of the circle's area, unlike a sector's. (b) A chord of a circle of radius 20 cm subtends 90°. Find the minor and major segment areas in terms of $\pi$. (c) Describe two checks you would apply to those answers.

Q116. (a) Derive the chord formula $2r\sin\frac{\theta}{2}$ and give the chord for each permitted angle. (b) Explain the error in using $\frac12 \times \text{chord} \times r$ for the triangle's area. (c) A chord of a circle of radius 6 cm subtends 60°. Find the minor segment's area in terms of $\pi$, and the segment's perimeter.