Question

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finding the area of the segment

find the area of the segment in each circle. round your answer to two decimal places. (use 3.14 for \\(\pi\\).)

1)
radius = 13 in, central angle = 150\\(^{\circ}\\)

2)
radius = 20 ft, central angle = 84\\(^{\circ}\\)

3)
radius = 6 yd, central angle = 77\\(^{\circ}\\)

4)
radius = 8 in, central angle = 130\\(^{\circ}\\)

5)
radius = 14 ft, central angle = 45\\(^{\circ}\\)

6)
radius = 9 yd, central angle = 110\\(^{\circ}\\)

Explanation:

Understand the segment area formula

To find the area of a circular segment, subtract the area of the central triangle from the area of the corresponding sector:

$$ A_{\text{segment}} = A_{\text{sector}} - A_{\text{triangle}} $$

The sector area is given by:

$$ A_{\text{sector}} = \frac{\theta}{360^\circ} \cdot \pi r^2 $$

The triangle area is given by:

$$ A_{\text{triangle}} = \frac{1}{2} r^2 \sin(\theta) $$

We use \(\pi \approx 3.14\) as instructed.

Calculate questions 1 and 2

For Question 1:

• Radius \(r = 13\text{ in}\), central angle \(\theta = 150^\circ\).
• Sector area:

$$ A_{\text{sector}} = \frac{150}{360} \cdot 3.14 \cdot 13^2 \approx 221.01 $$

• Triangle area:

$$ A_{\text{triangle}} = \frac{1}{2} \cdot 13^2 \cdot \sin(150^\circ) = 84.50 $$

• Segment area:

$$ A_{\text{segment}} = 221.01 - 84.50 = 136.51\text{ in}^2 $$

For Question 2:

• Radius \(r = 20\text{ ft}\), central angle \(\theta = 84^\circ\).
• Sector area:

$$ A_{\text{sector}} = \frac{84}{360} \cdot 3.14 \cdot 20^2 \approx 293.07 $$

• Triangle area:

$$ A_{\text{triangle}} = \frac{1}{2} \cdot 20^2 \cdot \sin(84^\circ) \approx 198.90 $$

• Segment area:

$$ A_{\text{segment}} = 293.07 - 198.90 = 94.17\text{ ft}^2 $$

Calculate questions 3 and 4

For Question 3:

• Radius \(r = 6\text{ yd}\), central angle \(\theta = 77^\circ\).
• Sector area:

$$ A_{\text{sector}} = \frac{77}{360} \cdot 3.14 \cdot 6^2 \approx 24.18 $$

• Triangle area:

$$ A_{\text{triangle}} = \frac{1}{2} \cdot 6^2 \cdot \sin(77^\circ) \approx 17.54 $$

• Segment area:

$$ A_{\text{segment}} = 24.18 - 17.54 = 6.64\text{ yd}^2 $$

For Question 4:

• Radius \(r = 8\text{ in}\), central angle \(\theta = 130^\circ\).
• Sector area:

$$ A_{\text{sector}} = \frac{130}{360} \cdot 3.14 \cdot 8^2 \approx 72.57 $$

• Triangle area:

$$ A_{\text{triangle}} = \frac{1}{2} \cdot 8^2 \cdot \sin(130^\circ) \approx 24.51 $$

• Segment area:

$$ A_{\text{segment}} = 72.57 - 24.51 = 48.06\text{ in}^2 $$

Calculate questions 5 and 6

For Question 5:

• Radius \(r = 14\text{ ft}\), central angle \(\theta = 45^\circ\).
• Sector area:

$$ A_{\text{sector}} = \frac{45}{360} \cdot 3.14 \cdot 14^2 \approx 76.93 $$

• Triangle area:

$$ A_{\text{triangle}} = \frac{1}{2} \cdot 14^2 \cdot \sin(45^\circ) \approx 69.30 $$

• Segment area:

$$ A_{\text{segment}} = 76.93 - 69.30 = 7.63\text{ ft}^2 $$

For Question 6:

• Radius \(r = 9\text{ yd}\), central angle \(\theta = 110^\circ\).
• Sector area:

$$ A_{\text{sector}} = \frac{110}{360} \cdot 3.14 \cdot 9^2 \approx 77.72 $$

• Triangle area:

$$ A_{\text{triangle}} = \frac{1}{2} \cdot 9^2 \cdot \sin(110^\circ) \approx 38.06 $$

• Segment area:

$$ A_{\text{segment}} = 77.72 - 38.06 = 39.66\text{ yd}^2 $$

Answer:

No.	Answer
2	\(94.17\text{ ft}^2\)
3	\(6.64\text{ yd}^2\)
4	\(48.06\text{ in}^2\)
5	\(7.63\text{ ft}^2\)
6	\(39.66\text{ yd}^2\)