Area of the Segment Worksheets - Free Printable

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:
$A_{sector} = \frac{\theta}{360} \times \pi r^2$

The formula for the area of the triangle (when two sides and the included angle are known) is:
$A_{triangle} = \frac{1}{2} r^2 \sin(\theta)$

Therefore, the area of the segment is:
$A_{segment} = A_{sector} - A_{triangle} = \frac{\theta}{360} \times \pi r^2 - \frac{1}{2} r^2 \sin(\theta)$

Let's calculate the area for each problem:

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Problem 1:
- Radius $r = 19$ in
- Angle $\theta = 58^\circ$
- $\pi = 3.14$

$A_{sector} = \frac{58}{360} \times 3.14 \times 19^2 = \frac{58}{360} \times 3.14 \times 361 \approx 182.73$ in²

$A_{triangle} = \frac{1}{2} \times 19^2 \times \sin(58^\circ) = \frac{1}{2} \times 361 \times 0.8480 \approx 153.17$ in²

$A_{segment} = 182.73 - 153.17 = 29.56$ in²

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Problem 2:
- Radius $r = 20$ ft
- Angle $\theta = 84^\circ$

$A_{sector} = \frac{84}{360} \times 3.14 \times 20^2 = \frac{84}{360} \times 3.14 \times 400 \approx 293.22$ ft²

$A_{triangle} = \frac{1}{2} \times 20^2 \times \sin(84^\circ) = \frac{1}{2} \times 400 \times 0.9945 \approx 198.90$ ft²

$A_{segment} = 293.22 - 198.90 = 94.32$ ft²

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Problem 3:
- Radius $r = 6$ yd
- Angle $\theta = 77^\circ$

$A_{sector} = \frac{77}{360} \times 3.14 \times 6^2 = \frac{77}{360} \times 3.14 \times 36 \approx 24.19$ yd²

$A_{triangle} = \frac{1}{2} \times 6^2 \times \sin(77^\circ) = \frac{1}{2} \times 36 \times 0.9744 \approx 17.54$ yd²

$A_{segment} = 24.19 - 17.54 = 6.65$ yd²

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Problem 4:
- Radius $r = 8$ in
- Angle $\theta = 150^\circ$

$A_{sector} = \frac{150}{360} \times 3.14 \times 8^2 = \frac{150}{360} \times 3.14 \times 64 \approx 83.73$ in²

$A_{triangle} = \frac{1}{2} \times 8^2 \times \sin(150^\circ) = \frac{1}{2} \times 64 \times 0.5 = 16.00$ in²

$A_{segment} = 83.73 - 16.00 = 67.73$ in²

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Problem 5:
- Radius $r = 14$ ft
- Angle $\theta = 65^\circ$

$A_{sector} = \frac{65}{360} \times 3.14 \times 14^2 = \frac{65}{360} \times 3.14 \times 196 \approx 111.37$ ft²

$A_{triangle} = \frac{1}{2} \times 14^2 \times \sin(65^\circ) = \frac{1}{2} \times 196 \times 0.9063 \approx 88.82$ ft²

$A_{segment} = 111.37 - 88.82 = 22.55$ ft²

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Problem 6:
- Radius $r = 9$ yd
- Angle $\theta = 110^\circ$

$A_{sector} = \frac{110}{360} \times 3.14 \times 9^2 = \frac{110}{360} \times 3.14 \times 81 \approx 77.75$ yd²

$A_{triangle} = \frac{1}{2} \times 9^2 \times \sin(110^\circ) = \frac{1}{2} \times 81 \times 0.9397 \approx 38.06$ yd²

$A_{segment} = 77.75 - 38.06 = 39.69$ yd²

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Final Answers:

1) 29.56 in²
2) 94.32 ft²
3) 6.65 yd²
4) 67.73 in²
5) 22.55 ft²
6) 39.69 yd²

All answers are rounded to two decimal places as requested.