Welcome to Part 1 of our new 8-part series on Chapter 11, Areas Related to Circles. This post contains the top 25 Multiple Choice Questions (MCQs) to help you master finding area of sectors, segments, and combinations of plane figures.
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Top 25 MCQs - Areas Related to Circles
Question 1: Sector Area
The area of a sector of a circle with radius 6 cm if the angle of the sector is 60° is:
= (60/360) × (22/7) × 6 × 6
= (1/6) × (22/7) × 36
= 6 × 22/7 = 132/7 cm².
Question 2: Quadrant Area
The area of a quadrant of a circle whose circumference is 22 cm is:
2 × (22/7) × r = 22 => r = 7/2 cm.
Area of quadrant = (1/4)πr²
= (1/4) × (22/7) × (7/2) × (7/2)
= 77/8 cm².
Question 3: Arc Length
If the perimeter of a circle is equal to that of a square, then the ratio of their areas is:
Ratio of Areas = (πr²) / (a²)
= πr² / (πr/2)² = πr² / (π²r²/4) = 4/π
= 4 / (22/7) = 28/22 = 14/11.
Question 4: Minute Hand Area
The length of the minute hand of a clock is 14 cm. The area swept by the minute hand in 5 minutes is:
Area = (30/360) × (22/7) × 14 × 14
= (1/12) × 22 × 2 × 14
= 154/3 cm².
Question 5: Sector Perimeter
The perimeter of a sector of a circle of radius 10.5 cm is 44 cm. What is the angle of the sector?
44 = L + 2(10.5) => L = 44 - 21 = 23 cm.
L = (θ/360) × 2πr
23 = (θ/360) × 2 × (22/7) × 10.5
Solving gives θ ≈ 125° (Wait, let's recheck. L=23? Wait, 44 = L + 21. L=23. 23 = θ/360 * 66. θ = 23*360/66 = 125.45. Maybe options are approx or standard. Let's re-check with standard values. Let's try θ=120. L = 1/3 * 66 = 22. P = 22+21=43. Close. Let's assume question meant arc length. Wait. Let's assume question meant 43 cm. No.
Let's try a different standard problem. "Perimeter of sector of angle 60 and radius 7". P = 7.33 + 14 = 21.33.
Let's take a simpler PYQ: "If the perimeter of a semi-circular protractor is 36 cm, find its diameter."
Perimeter = πr + 2r = r(π+2) = 36. r(36/7)=36. r=7. d=14. Correct.
Replacement for Q5: The perimeter of a semicircular protractor whose radius is 'r' is:
(a) πr + 2r
(b) πr + r
(c) πr
(d) πr + 2πr
Question 5: Semicircle Perimeter
The perimeter of a semicircular protractor whose radius is 'r' is:
Question 6: Ring Area
The area of a circular ring formed by two concentric circles whose radii are 5.7 cm and 4.3 cm is:
= (22/7)(5.7 + 4.3)(5.7 - 4.3)
= (22/7)(10)(1.4)
= (22/7)(14) = 44 cm².
Question 7: Wheel Revolutions
The number of revolutions made by a circular wheel of radius 0.7 m in rolling a distance of 176 m is:
Number of revolutions = Total Distance / Circumference
= 176 / 4.4 = 40.
Question 8: Square Inscribed
The area of a square that can be inscribed in a circle of radius 8 cm is:
Side × √2 = 16 => Side = 16/√2 = 8√2 cm.
Area = Side² = (8√2)² = 64 × 2 = 128 cm².
Question 9: Area & Circumference
If the area of a circle is numerically equal to twice its circumference, then the diameter of the circle is:
r = 4.
Diameter = 2r = 8 units.
Question 10: Circle & Square
The radius of a circle whose circumference is equal to the sum of the circumferences of two circles of diameters 36 cm and 20 cm is:
Radii are 18 cm and 10 cm.
R = 18 + 10 = 28 cm.
Question 11: Largest Triangle
The area of the largest triangle that can be inscribed in a semi-circle of radius r is:
Area = 1/2 × Base × Height = 1/2 × 2r × r = r².
Question 12: Ratio of Areas
If the circumference of two circles are in the ratio 2:3, then the ratio of their areas is:
Ratio of areas = (r₁/r₂)² = (2/3)² = 4:9.
Question 13: Circle in Square
The area of the circle that can be inscribed in a square of side 6 cm is:
Radius = 3 cm.
Area = πr² = π(3)² = 9π cm².
Question 14: Path Area
A circular park has a path of uniform width around it. The difference between outer and inner circumferences is 132 m. The width of the path is:
2π(R - r) = 132.
(2 × 22/7) × Width = 132.
Width = 132 × 7 / 44 = 3 × 7 = 21 m.
Question 15: Sector Arc
Length of an arc of a sector of angle θ for a circle of radius r is:
Question 16: Quadrant Perimeter
The perimeter of a quadrant of a circle of radius 'r' is:
Two radii = 2r.
Total Perimeter = πr/2 + 2r.
Question 17: Area Difference
The area of a circle is 220 cm². The area of a square inscribed in it is:
Diameter = Diagonal of square = 2r.
Side² + Side² = (2r)². 2a² = 4r². a² = 2r².
Area of square = a² = 2(70) = 140 cm².
Question 18: Wire Shape
A wire can be bent in the form of a circle of radius 56 cm. If it is bent in the form of a square, then its area will be:
Perimeter of square = 4a = 352 => a = 88 cm.
Area = a² = 88 × 88 = 7744 cm².
Question 19: Area Change
If the circumference of a circle increases from 2π to 4π, then its area ___ the original area.
Area is proportional to r². If r doubles, area becomes (2)² = 4 times.
Question 20: Radius from Area Sum
The diameter of a circle whose area is equal to the sum of the areas of the two circles of radii 24 cm and 7 cm is:
R² = 576 + 49 = 625.
R = 25 cm.
Diameter = 2R = 50 cm.
Question 21: Wheel Distance
The distance covered by a circular wheel of diameter 'd' in 500 revolutions is:
Distance in 500 revolutions = 500πd.
Question 22: Major Sector Area
Area of the major sector of a circle of radius 35 cm and central angle 90° is:
Area = (270/360) × πr² = (3/4) × (22/7) × 35 × 35
= (3/4) × 22 × 5 × 35
= 2887.5 cm².
Question 23: Segment Area Formula
Area of a segment of a circle = Area of corresponding sector - ________.
Question 24: Clock Hand
The angle described by the minute hand in 20 minutes is:
In 20 minutes: 20 × 6° = 120°.
Question 25: Max Triangle Area
The area of the largest triangle that can be inscribed in a circle of radius R is:
Side of inscribed equilateral triangle (a) = R√3.
Area = (√3/4)a² = (√3/4)(R√3)² = (√3/4)(3R²) = (3√3/4)R².
