p12ArcSec - Arc Length and Sector Area Practice.ia2 - AbbyNet - Free Printable

Let’s solve each problem step by step.

We are finding the area of a sector of a circle.
The formula is:

> Sector Area = (θ / 360) × π × r²

Where:
- θ = central angle in degrees
- r = radius of the circle
- π ≈ 3.1416 (we’ll use this unless told otherwise)

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Problem 1:

Radius = 5 m, Angle = 270°

Area = (270/360) × π × 5²
= (3/4) × π × 25
= (75/4) × π
≈ 18.75 × 3.1416 ≈ 58.90 m²

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Problem 2:

Radius = 8 ft, Angle = 135°

Area = (135/360) × π × 8²
= (3/8) × π × 64
= (192/8) × π = 24 × π
≈ 24 × 3.1416 ≈ 75.40 ft²

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Problem 3:

Radius = 19 m, Angle = 230°

Area = (230/360) × π × 19²
= (23/36) × π × 361
First: 23 × 361 = let’s compute that → 23×360=8280, plus 23×1=23 → 8303
So: 8303 / 36 × π ≈ 230.6389 × π
≈ 230.6389 × 3.1416 ≈ 724.58 m²

*(Double-check: 230/360 = 0.63888..., 19²=361, 0.63888×361≈230.638, times π≈724.58 — correct)*

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Problem 4:

Radius = 13 mi, Angle = 240°

Area = (240/360) × π × 13²
= (2/3) × π × 169
= (338/3) × π ≈ 112.6667 × π
≈ 112.6667 × 3.1416 ≈ 353.95 mi²

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Problem 5:

Radius = 14 mi, Angle = 42°

Area = (42/360) × π × 14²
= (7/60) × π × 196
= (1372 / 60) × π ≈ 22.8667 × π
≈ 22.8667 × 3.1416 ≈ 71.84 mi²

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Problem 6:

Radius = 16 yd, Angle = 255°

Area = (255/360) × π × 16²
Simplify fraction: divide numerator and denominator by 15 → 17/24
So: (17/24) × π × 256
= (17 × 256) / 24 × π
256 ÷ 8 = 32, 24 ÷ 8 = 3 → better to compute directly:
17 × 256 = 4352
4352 / 24 = 181.333...
Then: 181.333 × π ≈ 181.333 × 3.1416 ≈ 569.65 yd²

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Problem 7:

Radius = 10 in, Angle = 45°

Area = (45/360) × π × 10²
= (1/8) × π × 100
= 12.5 × π ≈ 12.5 × 3.1416 ≈ 39.27 in²

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Problem 8:

Radius = 11 km, Angle = 210°

Area = (210/360) × π × 11²
= (7/12) × π × 121
= (847 / 12) × π ≈ 70.5833 × π
≈ 70.5833 × 3.1416 ≈ 221.74 km²

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Problem 9:

Radius = 14 in, Angle = 90°

Area = (90/360) × π × 14²
= (1/4) × π × 196
= 49 × π ≈ 49 × 3.1416 ≈ 153.94 in²

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Problem 10:

Radius = 7 m, Angle = 150°

Area = (150/360) × π × 7²
= (5/12) × π × 49
= (245 / 12) × π ≈ 20.4167 × π
≈ 20.4167 × 3.1416 ≈ 64.14 m²

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Problem 11:

r = 8 in, θ = 150°

Area = (150/360) × π × 8²
= (5/12) × π × 64
= (320 / 12) × π ≈ 26.6667 × π
≈ 26.6667 × 3.1416 ≈ 83.78 in²

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Problem 12:

r = 10 in, θ = 135°

Area = (135/360) × π × 10²
= (3/8) × π × 100
= (300 / 8) × π = 37.5 × π
≈ 37.5 × 3.1416 ≈ 117.81 in²

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Problem 13:

r = 4 ft, θ = 45°

Area = (45/360) × π × 4²
= (1/8) × π × 16
= 2 × π ≈ 2 × 3.1416 ≈ 6.28 ft²

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Problem 14:

r = 16 km, θ = 150°

Area = (150/360) × π × 16²
= (5/12) × π × 256
= (1280 / 12) × π ≈ 106.6667 × π
≈ 106.6667 × 3.1416 ≈ 335.10 km²

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Final Answer:
1) 58.90 m²
2) 75.40 ft²
3) 724.58 m²
4) 353.95 mi²
5) 71.84 mi²
6) 569.65 yd²
7) 39.27 in²
8) 221.74 km²
9) 153.94 in²
10) 64.14 m²
11) 83.78 in²
12) 117.81 in²
13) 6.28 ft²
14) 335.10 km²