Areas Related to Circles — Karnataka (SSLC) Class 10 Mathematics Solutions (Free)

Step 1: Area of a sector = (θ/360°) × πr² Step 2: Here, θ = 60°, r = 7 cm, π = 22/7 Step 3: Area = (60°/360°) × (22/7) × 7² Step 4: Area = (1/6) × (22/7) × 49 Step 5: Area = (1/6) × 22 × 7 Step 6: Area = (1/6) × 154 Step 7: Area = 154/6 = 77/3 cm² Step 8: Area ≈ 25.67 cm² Final Answer: 77/3 cm² or approximately 25.67 cm²

Step 1: Perimeter of a sector = 2r + arc length Step 2: Here, r = 10.5 cm, arc length = 11 cm Step 3: Perimeter = 2(10.5) + 11 Step 4: Perimeter = 21 + 11 Step 5: Perimeter = 32 cm Final Answer: 32 cm

Step 1: Area of segment = Area of sector − Area of triangle Step 2: Area of sector = (θ/360°) × πr² = (90°/360°) × (22/7) × 14² Step 3: Area of sector = (1/4) × (22/7) × 196 Step 4: Area of sector = (1/4) × 22 × 28 Step 5: Area of sector = (1/4) × 616 = 154 cm² Step 6: Area of triangle = (1/2) × r² × sin(θ) = (1/2) × 14² × sin(90°) Step 7: Area of triangle = (1/2) × 196 × 1 = 98 cm² Step 8: Area of segment = 154 − 98 = 56 cm² Final Answer: 56 cm²

Step 1: Area of sector 1 = (120°/360°) × πr² Step 2: Area of sector 1 = (1/3) × (22/7) × 6² Step 3: Area of sector 1 = (1/3) × (22/7) × 36 Step 4: Area of sector 1 = (1/3) × 22 × 36/7 Step 5: Area of sector 1 = (22 × 12)/7 = 264/7 cm² Step 6: Area of sector 2 = (150°/360°) × πr² Step 7: Area of sector 2 = (5/12) × (22/7) × 36 Step 8: Area of sector 2 = (5/12) × 22 × 36/7 Step 9: Area of sector 2 = (5 × 22 × 3)/7 = 330/7 cm² Step 10: Difference = 330/7 − 264/7 = 66/7 ≈ 9.43 cm² Final Answer: 66/7 cm² or approximately 9.43 cm²

Step 1: First, find the central angle using the chord length Step 2: For a chord, chord = 2r × sin(θ/2) Step 3: 6 = 2(5) × sin(θ/2) Step 4: sin(θ/2) = 6/10 = 0.6 Step 5: θ/2 = 36.87°, so θ ≈ 73.74° ≈ 74° (approximately) Step 6: Using cos formula: cos(θ/2) = √(1 − 0.36) = √0.64 = 0.8 Step 7: Area of sector = (θ/360°) × πr² = (74°/360°) × (22/7) × 25 Step 8: Area of sector ≈ 16.09 cm² Step 9: Area of triangle = (1/2) × r² × sin(θ) = (1/2) × 25 × sin(74°) Step 10: Area of triangle ≈ (1/2) × 25 × 0.96 ≈ 12 cm² Step 11: Area of segment = 16.09 − 12 ≈ 4.09 cm² Final Answer: Approximately 4.1 cm²

Step 1: Arc length = (θ/360°) × 2πr Step 2: 22 = (θ/360°) × 2 × (22/7) × 21 Step 3: 22 = (θ/360°) × 2 × 22 × 3 Step 4: 22 = (θ/360°) × 132 Step 5: θ/360° = 22/132 = 1/6 Step 6: θ = 60° Step 7: Area of sector = (θ/360°) × πr² = (60°/360°) × (22/7) × 21² Step 8: Area of sector = (1/6) × (22/7) × 441 Step 9: Area of sector = (1/6) × 22 × 63 Step 10: Area of sector = (1/6) × 1386 = 231 cm² Final Answer: 231 cm²

Step 1: Area of ring = Area of outer circle − Area of inner circle Step 2: Area of ring = πR² − πr² Step 3: Here, R = 5 cm (outer radius), r = 3 cm (inner radius) Step 4: Area of ring = π(R² − r²) = (22/7)(5² − 3²) Step 5: Area of ring = (22/7)(25 − 9) Step 6: Area of ring = (22/7) × 16 Step 7: Area of ring = 352/7 ≈ 50.29 cm² Final Answer: 352/7 cm² or approximately 50.29 cm²

Step 1: Arc length = (θ/360°) × 2πr Step 2: Arc length = (54°/360°) × 2 × 3.14 × 10 Step 3: Arc length = (54/360) × 62.8 Step 4: Arc length = 0.15 × 62.8 = 9.42 cm Step 5: Area of sector = (θ/360°) × πr² Step 6: Area of sector = (54°/360°) × 3.14 × 10² Step 7: Area of sector = (54/360) × 3.14 × 100 Step 8: Area of sector = 0.15 × 314 = 47.1 cm² Final Answer: Arc length = 9.42 cm, Area = 47.1 cm²

Step 1: Area between circles = π(R² − r²) Step 2: Here, R = 7 cm, r = 5 cm Step 3: Area = (22/7)(7² − 5²) Step 4: Area = (22/7)(49 − 25) Step 5: Area = (22/7) × 24 Step 6: Area = 22 × 24/7 = 528/7 ≈ 75.43 cm² Step 7: Perimeter of outer circle = 2πR = 2 × (22/7) × 7 Step 8: Perimeter = 2 × 22 = 44 cm Final Answer: Area between circles = 528/7 cm² ≈ 75.43 cm², Perimeter of outer circle = 44 cm