Basic Geometric Concepts - Problems on Circles

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Q1. What is the area of a sector of a circle with a radius of 10 cm and a central angle of 60 degrees?

Solution:

Area of sector = (θ/360) * πr²; = (60/360) * π(10)² = (1/6) * 100π = 50π/3 cm².

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Q2. What is the length of the diameter of a circle with an area of 50π cm²?

Solution:

Area = πr²; 50π = πr²; r² = 50; r = √50; Diameter = 2r = 2√50 = 10 cm.

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Q3. What is the distance between the center of a circle at (2, 3) and a point on the circle at (5, 7)?

Solution:

Distance = √((5-2)² + (7-3)²) = √(3² + 4²) = √(9 + 16) = √25 = 5.

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Q4. If a circle has a radius of 9 cm, what is the area of the circle?

Solution:

Area = πr² = π(9)² = 81π cm².

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Q5. If a circle has a radius of 2 m, what is the area of the circle?

Solution:

Area = πr² = π(2)² = 4π m².

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Q6. A circle is inscribed in a triangle with sides 6 cm, 8 cm, and 10 cm. What is the radius of the inscribed circle?

Solution:

Semi-perimeter = (6+8+10)/2 = 12 cm; Area = √(12(12-6)(12-8)(12-10)) = 24 cm²; Radius = Area/semi-perimeter = 24/12 = 2 cm.

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Q7. If two circles have radii of 3 cm and 6 cm, what is the ratio of their areas?

Solution:

Area ratio = (r1²):(r2²) = (3²):(6²) = 9:36 = 1:4.

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Q8. What is the distance between the center of a circle with radius 5 cm and a point on its circumference?

Solution:

The distance from the center to a point on the circumference is equal to the radius, which is 5 cm.

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Q9. If a circle has an area of 36π cm², what is its radius?

Solution:

Area = πr²; 36π = πr²; r² = 36; r = 6 cm.

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Q10. A circle is inscribed in a triangle with sides 8 cm, 15 cm, and 17 cm. What is the radius of the inscribed circle?

Solution:

Semi-perimeter = (8 + 15 + 17)/2 = 20 cm. Area = 60 cm². Radius = Area/semi-perimeter = 60/20 = 3 cm.