Step 1: Understand what we're finding: Arc length is a portion of the circumference

Step 2: Use the arc length formula: Arc length = (θ/360°) × 2πr where θ is the central angle in degrees

Step 3: Substitute values: Arc length = (60°/360°) × 2π(6) Arc length = (1/6) × 12π Arc length = 2π cm

Step 4: Approximate (optional): 2π ≈ 2 × 3.14159 ≈ 6.28 cm

Step 5: Verify the logic: 60° is 1/6 of 360° (full circle) So arc is 1/6 of circumference C = 2π(6) = 12π Arc = 12π/6 = 2π ✓

Answer: Arc length = 2π cm (≈ 6.28 cm)

Use the arc length formula: $L = \frac{\theta}{360°} \times 2\pi r$

$L = \frac{60}{360} \times 2\pi(6)$

$L = \frac{1}{6} \times 12\pi$

$L = 2\pi$

Answer: $2\pi$ (or approximately 6.28) units

Step 1: Use the sector area formula: Sector area = (θ/360°) × πr²

Step 2: Substitute values: Sector area = (90°/360°) × π(8)² Sector area = (1/4) × 64π Sector area = 16π cm²

Step 3: Approximate (optional): 16π ≈ 50.27 cm²

Step 4: Verify: 90° is 1/4 of 360° Total circle area = π(8)² = 64π Sector = 64π/4 = 16π ✓

Answer: Sector area = 16π cm² (≈ 50.27 cm²)

Use the sector area formula: $A = \frac{\theta}{360°} \times \pi r^2$

$A = \frac{135}{360} \times \pi(8)^2$

$A = \frac{3}{8} \times 64\pi$

$A = 24\pi$

Answer: $24\pi$ (or approximately 75.4) square units

Step 1: Use arc length formula: Arc length = (θ/360°) × 2πr

Step 2: Substitute known values: 10π = (120°/360°) × 2πr 10π = (1/3) × 2πr 10π = (2π/3)r

Step 3: Solve for r: 10π = (2π/3)r 10π × (3/2π) = r 30π/2π = r r = 15 cm

Step 4: Verify: Arc = (120°/360°) × 2π(15) Arc = (1/3) × 30π Arc = 10π ✓

Answer: The radius is 15 cm

Step 1: Use sector area formula: Sector area = (θ/360°) × πr²

Step 2: Substitute known values: 27π = (135°/360°) × πr²

Step 3: Simplify the fraction: 135°/360° = 3/8 27π = (3/8) × πr²

Step 4: Solve for r²: 27π = (3π/8)r² 27π × (8/3π) = r² 216π/3π = r² 72 = r²

Step 5: Solve for r: r = √72 r = √(36 × 2) r = 6√2 meters

Step 6: Verify: Sector area = (3/8) × π(6√2)² = (3/8) × π(72) = (3/8) × 72π = 27π ✓

Answer: The radius is 6√2 meters (≈ 8.49 m)

Step 1: Find the central angle using arc length

$L = \frac{\theta}{360°} \times 2\pi r$

$5\pi = \frac{\theta}{360} \times 2\pi(10)$

$5\pi = \frac{\theta}{360} \times 20\pi$

$5 = \frac{\theta}{360} \times 20$

$5 = \frac{20\theta}{360}$

$\theta = 90°$

Step 2: Find the sector area

$A = \frac{90}{360} \times \pi(10)^2$

$A = \frac{1}{4} \times 100\pi = 25\pi$

Answer: Central angle = $90°$, Area = $25\pi$ square units

Step 1: Understand the setup: Arc length = 18 inches Radius (pendulum length) = 24 inches Need to find the central angle θ

Step 2: Use arc length formula to find θ: Arc length = (θ/360°) × 2πr 18 = (θ/360°) × 2π(24) 18 = (θ/360°) × 48π

Step 3: Solve for θ: 18 × 360° = θ × 48π 6480° = 48πθ θ = 6480°/(48π) θ = 135°/π

Step 4: Calculate θ in degrees: θ = 135°/π ≈ 135°/3.14159 ≈ 42.97°

Step 5: Find sector area: Sector area = (θ/360°) × πr²

Using exact value θ = 135°/π: Sector area = (135°/π / 360°) × π(24)² Sector area = (135°/(360°π)) × 576π Sector area = (135° × 576π)/(360°π) Sector area = (135° × 576)/(360°) Sector area = 77760°/360° Sector area = 216 square inches

Step 6: Alternative method for area: Area = (1/2) × arc length × radius Area = (1/2) × 18 × 24 Area = 216 square inches ✓

Answer: (a) Central angle ≈ 42.97° (or exactly 135°/π) (b) Sector area = 216 square inches