Chapter 11 · Class 10 CBSE · MCQ Test
Areas Related to Circles MCQ Test — Class 10 CBSE
Practice 10 multiple-choice questions with instant answer reveal and explanations.
Areas Related to Circles — MCQ Questions
1Which of the following statements correctly defines a 'sector' of a circle?
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Answer: The region enclosed by two radii and their corresponding arc.
Hint: Recall the fundamental definition of a sector. It involves the center of the circle and its boundary.
Solution:
A sector of a circle is formed by two radii originating from the center of the circle and the arc connecting their endpoints on the circle's circumference.
Option A describes a segment, not a sector. Option B is incorrect as it mentions a diameter, which is not a defining element for a sector. Option D is also incorrect.
Thus, the correct definition involves two radii and the arc.
2A student calculates the area of a sector with radius 'r' and angle 'θ' as (θ/360°) × πr². Their friend says this formula is only for minor sectors. Is the friend correct?
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Answer: No, the formula applies to both minor and major sectors.
Hint: Consider what 'θ' represents in the formula. It's the angle *subtended by the arc* at the center, regardless of whether it's acute or reflex.
Solution:
The formula for the area of a sector, Area = (θ/360°) × πr², is a general formula.
If 'θ' represents the angle of the minor sector (e.g., 60°), it calculates the minor sector's area.
If 'θ' represents the angle of the major sector (i.e., 360° - angle of minor sector, e.g., 300°), it calculates the major sector's area.
Therefore, the formula is universally applicable, provided the correct angle 'θ' for the desired sector is used.
3Ravi wants to find the length of the arc of a sector with radius 7 cm and angle 90°. He applies the formula: Arc Length = (90/360) × π(7)². What mistake did Ravi make?
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Answer: He used the formula for the area of a sector instead of arc length.
Hint: Carefully examine the formula Ravi used. Does it calculate a length or an area?
Solution:
The formula for the length of an arc is (θ/360°) × 2πr.
Ravi used (θ/360°) × πr², which is the formula for the area of a sector, not the length of an arc.
The angle (90°) and radius (7 cm) are correctly identified for the problem, but the base formula applied is wrong.
Therefore, the mistake is in using the area formula instead of the arc length formula.
4What is the area of a quadrant of a circle with a radius of 14 cm? (Use π = 22/7)
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Answer: 154 cm²
Hint: A quadrant is a sector formed by an angle of 90° or 1/4th of a circle. Use the area of a circle formula.
Solution:
A quadrant of a circle is a sector with a central angle of 90°.
The area of a circle is given by A = πr².
The area of a quadrant is (1/4) × πr² or (90/360) × πr².
Substituting r = 14 cm and π = 22/7: Area = (1/4) × (22/7) × (14)² = (1/4) × (22/7) × 196 = (1/4) × 22 × 28 = 22 × 7 = 154 cm².
5The area of a segment of a circle is calculated by subtracting the area of the corresponding __________ from the area of the corresponding __________.
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Answer: triangle, sector
Hint: Visualise a segment. It's the region bounded by an arc and a chord. How can you find its area using a sector?
Solution:
A segment of a circle is the region bounded by an arc and its corresponding chord.
To find the area of this segment, we first consider the sector formed by the same arc and the two radii to its endpoints.
From this sector, we subtract the area of the triangle formed by the two radii and the chord.
Therefore, the area of a segment = Area of sector - Area of corresponding triangle.
6Consider a major sector of a circle with radius 'r' and minor angle 'θ'. Which of the following is a valid method to find the area of the major sector?
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Answer: Calculate ((360° - θ)/360°) × πr².
Hint: The angle for the major sector is the reflex angle. Use the general sector area formula with this angle.
Solution:
The major sector is defined by the reflex angle at the center, which is (360° - θ) if θ is the angle of the minor sector.
The formula for the area of any sector is (Angle/360°) × πr².
Substituting the angle for the major sector, we get Area = ((360° - θ)/360°) × πr².
Option A is also mathematically correct but involves two steps. Option B is a direct application. Options C and D are generally incorrect for finding a major sector's area.
7If the area of a sector of a circle with radius 6 cm is 18π cm², what is the measure of the central angle of the sector?
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Answer: 180°
Hint: Use the formula for the area of a sector and solve for the unknown angle 'θ'.
Solution:
The formula for the area of a sector is A = (θ/360°) × πr².
Given A = 18π cm² and r = 6 cm. Substitute these values into the formula: 18π = (θ/360°) × π(6)².
Simplify the equation: 18π = (θ/360°) × 36π. Divide both sides by π: 18 = (θ/360°) × 36.
Solve for θ: θ = (18 × 360) / 36 = 18 × 10 = 180°.
8A clock's minute hand is 10 cm long. How far does the tip of the minute hand move in 30 minutes? (Use π = 3.14)
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Answer: 31.4 cm
Hint: The minute hand completes a full circle (360°) in 60 minutes. Determine the angle covered in 30 minutes and then use the arc length formula.
Solution:
In 60 minutes, the minute hand completes a full circle, covering 360°.
In 30 minutes, it covers half a circle, which is 30/60 × 360° = 180°.
The length of the minute hand is the radius (r = 10 cm). The distance moved by the tip is the arc length.
Arc Length = (θ/360°) × 2πr = (180°/360°) × 2 × 3.14 × 10 = (1/2) × 2 × 3.14 × 10 = 3.14 × 10 = 31.4 cm.
9A square ABCD has a side length of 7 cm. A quadrant of a circle with center A and radius AD is drawn inside the square. What is the area of the shaded region, which is the part of the square *outside* the quadrant? (Use π = 22/7)
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Answer: 10.5 cm²
Hint: The shaded area is the difference between the area of the square and the area of the quadrant.
Solution:
The side length of the square is 7 cm. So, Area of square = side × side = 7 cm × 7 cm = 49 cm².
The quadrant has its center at A and radius AD, which is the side length of the square, so r = 7 cm.
Area of the quadrant = (1/4) × πr² = (1/4) × (22/7) × (7)² = (1/4) × (22/7) × 49 = (1/4) × 22 × 7 = (1/4) × 154 = 38.5 cm².
Area of the shaded region = Area of square - Area of quadrant = 49 cm² - 38.5 cm² = 10.5 cm².
10If the radius of a circle is doubled, how does its area change?
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Answer: The area becomes four times the original area.
Hint: Write down the formula for the area of a circle and substitute 2r for r. Compare the new area to the original area.
Solution:
Let the original radius be 'r'. The original area of the circle is A₁ = πr².
If the radius is doubled, the new radius becomes '2r'.
The new area of the circle is A₂ = π(2r)² = π(4r²) = 4πr².
Comparing A₂ with A₁: A₂ = 4(πr²) = 4A₁. So, the area becomes four times the original area.
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Tips for Areas Related to Circles MCQs
- 1Read each question carefully and identify what is being asked before looking at the options.
- 2Try to solve the problem mentally or on paper first, then match your answer with the options.
- 3Use elimination — rule out clearly wrong options to improve your chances even when unsure.
- 4Check units, signs, and edge cases — these are common traps in Areas Related to Circles MCQs.
- 5Review your mistakes after completing the test to build lasting understanding.
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