SUMMARY: This chapter focuses on calculating the areas of circles and related figures such as sectors and segments. KEY TOPICS: area of a circle, perimeter of a circle, area of a sector, area of a segment, problems on finding areas, application of areas in real-life situations, conversion between units, use of π (pi), solving problems involving combinations of plane figures, examples and exercises.
What is the area of a circle with a radius of 7 cm? (Use π = 22/7)
A154 cm²
B144 cm²
C138 cm²
D160 cm²
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Correct answer: Option 1 — 154 cm²
Q21 Mark
A sector of a circle has a central angle of 60 degrees and a radius of 10 cm. What is the area of the sector? (Use π = 3.14)
A52.33 cm²
B25.00 cm²
C31.42 cm²
D16.67 cm²
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Correct answer: Option 1 — 52.33 cm²
Q31 Mark
If the circumference of a circle is 31.4 cm, what is the radius of the circle? (Use π = 3.14)
A5 cm
B6 cm
C7 cm
D8 cm
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Correct answer: Option 1 — 5 cm
Q41 Mark
A circular garden has a diameter of 14 m. What is the area of the garden in square meters? (Use π = 3.14)
A153.86 m²
B154 m²
C150 m²
D140 m²
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Correct answer: Option 1 — 153.86 m²
Q51 Mark
A segment of a circle has a radius of 10 cm and a central angle of 90 degrees. What is the area of the segment? (Use π = 3.14)
A24.57 cm²
B15.71 cm²
C20.00 cm²
D30.00 cm²
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Correct answer: Option 2 — 15.71 cm²
Short Answer Questions5 questions
Q63 Marks
What is the formula to calculate the area of a circle?
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The area of a circle can be calculated using the formula A = πr², where A is the area and r is the radius of the circle.
Q73 Marks
How do you find the area of a sector of a circle?
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The area of a sector can be found using the formula A = (θ/360) × πr², where θ is the angle of the sector in degrees and r is the radius of the circle.
Q83 Marks
Calculate the area of a circle with a radius of 7 cm. Use π = 22/7 for your calculations.
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Using the formula A = πr², the area is A = (22/7) × (7)² = (22/7) × 49 = 154 cm².
Q93 Marks
Explain the difference between the area of a segment and the area of a sector in a circle.
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The area of a segment is the area of a sector minus the area of the triangle formed by the two radii and the chord, while the area of a sector is simply the portion of the circle defined by the angle at the center.
Q103 Marks
A circular garden has a diameter of 10 m. What is the perimeter (circumference) of the garden?
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The perimeter (circumference) of the garden can be calculated using the formula C = πd, where d is the diameter. Thus, C = π × 10 = 31.4 m (using π ≈ 3.14).
Long Answer Questions5 questions
Q116 Marks
A circular garden has a radius of 14 meters. Calculate the area of the garden and the perimeter of the garden. Also, find the area if a sector of the garden subtends a central angle of 60 degrees at the center. Show all your calculations.
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The area of the circular garden can be calculated using the formula A = πr², where r is the radius. Therefore, A = π(14)² = 196π square meters. The perimeter (circumference) is given by C = 2πr, so C = 2π(14) = 28π meters. For the sector with a central angle of 60 degrees, the area of the sector is A_sector = (θ/360) × πr² = (60/360) × π(14)² = (1/6) × 196π = 32.67π square meters. Thus, the area of the garden is 196π m², the perimeter is 28π m, and the area of the sector is approximately 32.67π m².
Q126 Marks
A segment of a circle has a radius of 10 cm and a central angle of 120 degrees. Calculate the area of the segment. Show your calculations step by step.
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First, calculate the area of the sector using the formula A_sector = (θ/360) × πr². Here, θ = 120 degrees and r = 10 cm. Thus, A_sector = (120/360) × π(10)² = (1/3) × 100π = 33.33π cm². Next, we need to find the area of the triangle formed by the two radii and the chord. The area of the triangle can be calculated using the formula A_triangle = (1/2) × r² × sin(θ). Therefore, A_triangle = (1/2) × (10)² × sin(120°) = (1/2) × 100 × (√3/2) = 25√3 cm². Finally, the area of the segment is A_segment = A_sector - A_triangle = 33.33π - 25√3 cm². Hence, the area of the segment is approximately 33.33π - 25√3 cm².
Q136 Marks
A circular swimming pool has a diameter of 20 meters. Calculate the area of the pool and the area of a circular path of width 2 meters surrounding the pool. Show all calculations.
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The radius of the swimming pool is half of the diameter, so r_pool = 20/2 = 10 meters. The area of the pool is A_pool = πr_pool² = π(10)² = 100π square meters. The radius of the outer circle (pool plus path) is r_outer = 10 + 2 = 12 meters. The area of the larger circle is A_outer = πr_outer² = π(12)² = 144π square meters. The area of the circular path is A_path = A_outer - A_pool = 144π - 100π = 44π square meters. Therefore, the area of the pool is 100π m² and the area of the circular path is 44π m².
Q146 Marks
A sector of a circle has a radius of 5 cm and a central angle of 90 degrees. Calculate the area of the sector and the length of the arc. Provide detailed calculations.
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To find the area of the sector, we use the formula A_sector = (θ/360) × πr². Here, θ = 90 degrees and r = 5 cm. Thus, A_sector = (90/360) × π(5)² = (1/4) × 25π = 6.25π cm². For the length of the arc, we use the formula L_arc = (θ/360) × 2πr. Therefore, L_arc = (90/360) × 2π(5) = (1/4) × 10π = 2.5π cm. Hence, the area of the sector is 6.25π cm² and the length of the arc is 2.5π cm.
Q156 Marks
A circular pizza has a radius of 8 inches. If a slice of the pizza represents a sector with a central angle of 45 degrees, calculate the area of the slice and the length of the crust (arc length). Show your calculations.
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First, calculate the area of the sector (slice) using the formula A_sector = (θ/360) × πr². Here, θ = 45 degrees and r = 8 inches. Thus, A_sector = (45/360) × π(8)² = (1/8) × 64π = 8π square inches. Next, calculate the length of the crust (arc length) using L_arc = (θ/360) × 2πr. Therefore, L_arc = (45/360) × 2π(8) = (1/8) × 16π = 2π inches. Hence, the area of the slice is 8π in² and the length of the crust is 2π inches.
Assertion–Reason Questions5 questions
Q161 Mark
Assertion (A): The area of a circle is calculated using the formula A = πr².
Reason (R): The radius is the distance from the center of the circle to any point on its circumference.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q171 Mark
Assertion (A): The perimeter of a circle is also known as its circumference.
Reason (R): The circumference can be calculated using the formula C = 2πr.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q181 Mark
Assertion (A): The area of a sector is given by the formula A = (θ/360) × πr².
Reason (R): This formula applies only when θ is measured in radians.
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Correct answer: Option 3 —
A is true, but R is false.
Q191 Mark
Assertion (A): A segment of a circle can be defined as the area enclosed between a chord and the corresponding arc.
Reason (R): Segments can only be formed in circles with a radius greater than zero.
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Correct answer: Option 2 —
Both A and R are true, but R is not the correct explanation of A.
Q201 Mark
Assertion (A): To find the area of a circle, one must always use the value of π as 3.14.
Reason (R): The value of π can vary based on the precision required for calculations.
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Correct answer: Option 4 —
A is false, but R is true.
Statement-Based Questions5 questions
Q211 Mark
Statement 1: The area of a circle is calculated using the formula A = πr².
Statement 2: The perimeter of a circle is also known as its circumference and is calculated using the formula C = 2πr.
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Correct answer: Option 1 —
Both statements are true.
Q221 Mark
Statement 1: The area of a sector of a circle can be found using the formula A = (θ/360) × πr², where θ is the angle in degrees.
Statement 2: The area of a segment of a circle is always greater than the area of the corresponding sector.
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Correct answer: Option 3 —
Only Statement 2 is true.
Q231 Mark
Statement 1: To find the area of a segment, you must first calculate the area of the triangle formed by the radii and the chord.
Statement 2: The value of π is approximately equal to 3.14.
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Correct answer: Option 1 —
Both statements are true.
Q241 Mark
Statement 1: If the radius of a circle is doubled, the area of the circle becomes four times larger.
Statement 2: The circumference of a circle is directly proportional to its radius.
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Correct answer: Option 1 —
Both statements are true.
Q251 Mark
Statement 1: The area of a circle with a radius of 7 cm is 154 cm².
Statement 2: To convert the area from cm² to m², you divide by 10000.
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Correct answer: Option 2 —
Only Statement 1 is true.
