Areas Related to Circles — Karnataka (SSLC) Class 10 Mathematics Solutions (Free)
Free step-by-step Karnataka (SSLC) Class 10 Mathematics solutions for "Areas Related to Circles" — important questions with detailed answers, download PDF for board exam preparation.
TL;DR: Free step-by-step Karnataka (SSLC) Class 10 Mathematics solutions for "Areas Related to Circles" — important questions with detailed answers, download…
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Q1: A sector of a circle has radius 7 cm and central angle 60°. Find the area of the sector. (Use π = 22/7)
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
Q2: Find the perimeter of a sector with radius 10.5 cm and arc length 11 cm.
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
Q3: The radius of a circle is 14 cm. Find the area of a segment if the central angle of the corresponding sector is 90°. (Use π = 22/7)
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
Q4: A circle has two sectors with central angles 120° and 150°. If the radius is 6 cm, find the difference between their areas. (Use π = 22/7)
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…
Q5: Find the area of a segment of a circle with radius 5 cm and chord length 6 cm.
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.…
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