I will solve the problem of finding the area of the segment in each circle. The area of a segment is calculated by subtracting the area of the triangle from the area of the sector.
The formula for the area of a sector is:
Area of sector = (θ/360) × π × r²
The formula for the area of a triangle (when two sides and the included angle are known) is:
Area of triangle = (1/2) × r × r × sin(θ) = (1/2) × r² × sin(θ)
Therefore, the area of the segment is:
Area of segment = Area of sector - Area of triangle
Area of segment = (θ/360) × π × r² - (1/2) × r² × sin(θ)
Area of segment = r² × [(θ/360) × π - (1/2) × sin(θ)]
I will calculate this for each problem, using π = 3.14 as instructed.
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Problem 1:
- Radius (r) = 19 in
- Angle (θ) = 58°
Area of segment = 19² × [(58/360) × 3.14 - (1/2) × sin(58°)]
= 361 × [0.1611 × 3.14 - 0.5 × 0.8480]
= 361 × [0.5059 - 0.4240]
= 361 × 0.0819
≈ 29.57 in²
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Problem 2:
- Radius (r) = 20 ft
- Angle (θ) = 84°
Area of segment = 20² × [(84/360) × 3.14 - (1/2) × sin(84°)]
= 400 × [0.2333 × 3.14 - 0.5 × 0.9945]
= 400 × [0.7326 - 0.4973]
= 400 × 0.2353
≈ 94.12 ft²
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Problem 3:
- Radius (r) = 6 yd
- Angle (θ) = 77°
Area of segment = 6² × [(77/360) × 3.14 - (1/2) × sin(77°)]
= 36 × [0.2139 × 3.14 - 0.5 × 0.9744]
= 36 × [0.6716 - 0.4872]
= 36 × 0.1844
≈ 6.64 yd²
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Problem 4:
- Radius (r) = 8 in
- Angle (θ) = 150°
Area of segment = 8² × [(150/360) × 3.14 - (1/2) × sin(150°)]
= 64 × [0.4167 × 3.14 - 0.5 × 0.5]
= 64 × [1.3083 - 0.25]
= 64 × 1.0583
≈ 67.73 in²
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Problem 5:
- Radius (r) = 14 ft
- Angle (θ) = 65°
Area of segment = 14² × [(65/360) × 3.14 - (1/2) × sin(65°)]
= 196 × [0.1806 × 3.14 - 0.5 × 0.9063]
= 196 × [0.5671 - 0.4532]
= 196 × 0.1139
≈ 22.32 ft²
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Problem 6:
- Radius (r) = 9 yd
- Angle (θ) = 110°
Area of segment = 9² × [(110/360) × 3.14 - (1/2) × sin(110°)]
= 81 × [0.3056 × 3.14 - 0.5 × 0.9397]
= 81 × [0.9596 - 0.4698]
= 81 × 0.4898
≈ 39.67 yd²
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Final Answers:
1) 29.57 in²
2) 94.12 ft²
3) 6.64 yd²
4) 67.73 in²
5) 22.32 ft²
6) 39.67 yd²
Parent Tip: Review the logic above to help your child master the concept of area of sector worksheet.
