Question

find the area of the shaded segment of the circle.

the area of the shaded segment is \\(\square\\) \\(\text{ft}^2\\).
(round to the nearest tenth as needed.)

Explanation:

Step1: Find area of sector

The formula for the area of a sector is $\frac{\theta}{360^\circ} \times \pi r^2$, where $\theta = 80^\circ$ and $r = 3$ ft.
Substitute values: $\frac{80^\circ}{360^\circ} \times \pi \times 3^2 = \frac{2}{9} \times 9\pi = 2\pi \approx 6.283$ ft².

Step2: Find area of triangle

The formula for the area of a triangle with two sides $r$ and included angle $\theta$ is $\frac{1}{2} r^2 \sin\theta$.
Substitute $r = 3$, $\theta = 80^\circ$: $\frac{1}{2} \times 3^2 \times \sin80^\circ = \frac{9}{2} \times \sin80^\circ \approx 4.5 \times 0.9848 \approx 4.432$ ft².

Step3: Subtract triangle area from sector area

Segment area = Sector area - Triangle area.
$6.283 - 4.432 \approx 1.851 \approx 1.9$ ft² (rounded to nearest tenth).

Answer:

1.9