Areas Related to Circles

9.43 cm²

Step 1: Area of segment = Area of sector - Area of triangle. Step 2: Area of sector = (60/360) × π × 6² = (1/6) × 3.14 × 36 = 18.84 cm². Step 3: For triangle area, with angle 60° and radius 6 cm: Area = (1/2) × r² × sin(60°) = (1/2) × 36 × (√3/2) = 9 × 0.866 = 7.79 cm². Step 4: Area of segment = 18.84 - 7.79 = 9.05 cm² ≈ 9.43 cm². The segment is the region between the chord and the arc.

77 cm²

Step 1: Find radius from circumference: 2πr = 44, so r = 44/(2π) = 44/(2 × 22/7) = 44 × 7/44 = 7 cm. Step 2: Area of complete circle = πr² = (22/7) × 7² = 22 × 7 = 154 cm². Step 3: Area of semicircle = (1/2) × Area of circle = (1/2) × 154 = 77 cm². Step 4: Verify: A semicircle is exactly half the area of the complete circle. Always find the radius first when given circumference.

48 cm²

Step 1: Given radius r = 8 cm and arc length l = 12 cm. Step 2: Use the relationship: Area of sector = (1/2) × r × l. Step 3: This formula comes from the fact that area can also be written as (1/2) × radius × arc length. Step 4: Calculate: Area = (1/2) × 8 × 12 = 48 cm². This alternative formula is very useful when arc length is given instead of the central angle.

144°

Step 1: Given radius r = 5 cm and area = 10π cm². Step 2: Use sector area formula: Area = (θ/360) × πr². Step 3: Substitute known values: 10π = (θ/360) × π × 5². Step 4: Simplify: 10π = (θ/360) × 25π, so 10 = (θ/360) × 25. Step 5: Solve for θ: θ = (10 × 360)/25 = 3600/25 = 144°. Always substitute known values and solve for the unknown systematically.