Infinite Algebra 2 worksheet on arc length and sector area with 12 practice problems.

Let’s solve each problem step by step. We’ll use the arc length formula:

> Arc Length = (θ / 360°) × 2πr — when θ is in degrees
> Arc Length = r × θ — when θ is in radians

We’ll round to the nearest tenth for problems 1–8, and leave exact or simplified forms for 9–12 as instructed.

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Problem 1:

Circle with radius 11 ft, central angle 315°

Arc Length = (315/360) × 2π(11)
= (7/8) × 22π
= (154π)/8 = 19.25π ≈ 60.5 ft

✔ Check: 315/360 = 0.875; 0.875 × 2 × π × 11 ≈ 0.875 × 69.115 ≈ 60.5 → Correct

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Problem 2:

Radius 13 ft, angle 270°

Arc Length = (270/360) × 2π(13)
= (3/4) × 26π = (78π)/4 = 19.5π ≈ 61.3 ft

✔ Check: 270/360 = 0.75; 0.75 × 2 × π × 13 ≈ 0.75 × 81.68 ≈ 61.3 → Correct

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Problem 3:

Radius 16 ft, angle 2π/3 radians

Use radian formula: Arc Length = r × θ = 16 × (2π/3) = 32π/3 ≈ 33.5 ft

✔ Check: 32 × 3.1416 / 3 ≈ 100.53 / 3 ≈ 33.5 → Correct

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Problem 4:

Radius 13 m, angle π/6 radians

Arc Length = 13 × (π/6) = 13π/6 ≈ 6.8 m

✔ Check: 13 × 3.1416 / 6 ≈ 40.84 / 6 ≈ 6.8 → Correct

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Problem 5:

r = 18 cm, θ = 60°

Arc Length = (60/360) × 2π(18) = (1/6) × 36π = 6π ≈ 18.8 cm

✔ Check: 60/360 = 1/6; 1/6 × 2 × π × 18 = 6π ≈ 18.8 → Correct

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Problem 6:

r = 16 m, θ = 75°

Arc Length = (75/360) × 2π(16) = (5/24) × 32π = (160π)/24 = 20π/3 ≈ 20.9 m

✔ Check: 75/360 = 5/24; 5/24 × 32π = (160π)/24 = 20π/3 ≈ 20.94 → rounds to 20.9 → Correct

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Problem 7:

r = 9 ft, θ = 7π/4 radians

Arc Length = 9 × (7π/4) = 63π/4 ≈ 49.5 ft

✔ Check: 63 × 3.1416 / 4 ≈ 197.92 / 4 ≈ 49.48 → rounds to 49.5 → Correct

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Problem 8:

r = 14 ft, θ = 19π/12 radians

Arc Length = 14 × (19π/12) = (266π)/12 = 133π/6 ≈ 69.6 ft

✔ Check: 133 × 3.1416 / 6 ≈ 417.83 / 6 ≈ 69.64 → rounds to 69.6 → Correct

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Now problems 9–12: “Do not round” — so we leave answers in terms of π or simplified fractions.

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Problem 9:

Radius 8 cm, angle 315°

Arc Length = (315/360) × 2π(8) = (7/8) × 16π = 14π cm

✔ Simplified: 315/360 = 7/8; 7/8 × 16π = 14π → Exact answer

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Problem 10:

Radius 19 ft, angle 150°

Arc Length = (150/360) × 2π(19) = (5/12) × 38π = (190π)/12 = 95π/6 ft

✔ Simplify: 150/360 = 5/12; 5/12 × 38π = 190π/12 = 95π/6 → Reduced fraction

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Problem 11:

Radius 14 cm, angle π/2 radians

Arc Length = 14 × (π/2) = 7π cm

✔ Simple multiplication → 7π cm

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Problem 12:

Radius 19 ft, angle 5π/4 radians

Arc Length = 19 × (5π/4) = 95π/4 ft

✔ Direct multiplication → 95π/4 ft

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Final Answer:
1) 60.5 ft
2) 61.3 ft
3) 33.5 ft
4) 6.8 m
5) 18.8 cm
6) 20.9 m
7) 49.5 ft
8) 69.6 ft
9) 14π cm
10) 95π/6 ft
11) 7π cm
12) 95π/4 ft