Basic Geometric Concepts - Problems on Circles

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Q1. What is the diameter of a circle with an area of 50π cm²?

Solution:

Area = πr², so r² = 50, thus r = √50 ≈ 7.07 cm, diameter = 2r ≈ 10 cm.

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Q2. 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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Q3. What is the diameter of a circle with a circumference of 31.4 cm?

Solution:

Circumference = πd, so d = Circumference/π = 31.4/π ≈ 10 cm.

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Q4. If two circles have radii of 3 cm and 5 cm, what is the distance between their centers if they are externally tangent?

Solution:

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

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Q5. A circle is inscribed in a square. If the side of the square is 8 cm, what is the radius of the circle?

Solution:

The radius of the inscribed circle is half the side of the square, so r = 8/2 = 4 cm.

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Q6. What is the radius of a circle whose area is 50π cm²?

Solution:

Area = πr², so r² = 50, r = √50 = 7.07 cm.

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Q7. 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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Q8. 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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Q9. 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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Q10. 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.