Basic Geometric Concepts - Circle Problems for Exams

Basic Geometric Concepts - Problems on Circles - Applications

Q. A circle has a circumference of 31.4 cm. What is the radius of the circle?

A. 5 cm
B. 10 cm
C. 7 cm
D. 15 cm

Solution

Circumference = 2πr => 31.4 = 2πr => r = 31.4/(2π) = 5 cm.

Correct Answer: A — 5 cm

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Q. If a circle has a radius of 10 m, what is the area of a circle segment formed by a central angle of 120 degrees?

A. 50π m²
B. 100π m²
C. 25π m²
D. 75π m²

Solution

Area of segment = Area of sector - Area of triangle; Area of sector = (120/360) * π(10)² = (1/3) * 100π = 33.33π m²; Area of triangle = (1/2) * r² * sin(θ) = (1/2) * 10² * √3/2 = 25√3 m²; Area of segment = 33.33π - 25√3.

Correct Answer: B — 100π m²

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Q. If a triangle is inscribed in a circle of radius 10 cm, what is the maximum area of the triangle?

A. 50 cm²
B. 100 cm²
C. 75 cm²
D. 80 cm²

Solution

Maximum area = (1/2) * r² * sin(θ) = (1/2) * 10² * 1 = 50 cm².

Correct Answer: B — 100 cm²

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Q. Two circles have radii of 3 cm and 5 cm. What is the distance between their centers if they are externally tangent?

A. 2 cm
B. 8 cm
C. 5 cm
D. 3 cm

Solution

Distance = r1 + r2 = 3 + 5 = 8 cm.

Correct Answer: B — 8 cm

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Q. What is the area of a circle with a radius of 8 cm?

A. 64π cm²
B. 32π cm²
C. 16π cm²
D. 24π cm²

Solution

Area = πr² = π(8)² = 64π cm².

Correct Answer: A — 64π cm²

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Q. What is the area of a sector of a circle with a radius of 6 cm and a central angle of 60 degrees?

A. 6π cm²
B. 12π cm²
C. 3π cm²
D. 9π cm²

Solution

Area of sector = (θ/360) * πr² = (60/360) * π(6)² = 12π cm².

Correct Answer: B — 12π cm²

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Q. What is the diameter of a circle whose circumference is 31.4 cm?

A. 10 cm
B. 5 cm
C. 15.7 cm
D. 20 cm

Solution

Circumference = πd; 31.4 = πd; d = 31.4/π ≈ 10 cm.

Correct Answer: A — 10 cm

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Q. What is the length of the diameter of a circle if its area is 50π cm²?

A. 10 cm
B. 5 cm
C. 20 cm
D. 15 cm

Solution

Area = πr² = 50π. Thus, r² = 50, r = √50 = 5√2 cm. Diameter = 2r = 10 cm.

Correct Answer: A — 10 cm

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Q. What is the radius of a circle if its area is 36π cm²?

A. 6 cm
B. 12 cm
C. 18 cm
D. 9 cm

Solution

Area = πr², so r² = 36, thus r = 6 cm.

Correct Answer: A — 6 cm

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Q. What is the radius of a circle inscribed in a triangle with sides 7 cm, 8 cm, and 9 cm?

A. 4 cm
B. 3 cm
C. 5 cm
D. 6 cm

Solution

Semi-perimeter s = (7 + 8 + 9)/2 = 12 cm. Area = √[s(s-a)(s-b)(s-c)] = √[12(12-7)(12-8)(12-9)] = √[12*5*4*3] = 12√5 cm². Radius = Area/s = 12√5/12 = √5 cm.

Correct Answer: B — 3 cm

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Basic Geometric Concepts - Problems on Circles - Applications

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