# What is the area of the shaded segment?

## Mar 19, 2018

The area of the shaded segment is $\approx 0.5 {\text{ ft}}^{2}$

#### Explanation:

The formula for the area of a circle is $\pi {r}^{2}$. The radius of this circle is 3 feet. The area of the circle is ${3}^{2} \pi$, which is equal to $9 \pi$ ft$^ 2$. Since the angle of the arc that is shaded is 50 degrees, it is $\frac{50}{360}$ of the whole circle. The area of that specific segment plus the area of the isosceles triangle is the area of a sector of the circle. The latter would be the total area of the circle multiplied by the portion of the circle that segment takes up.
The area is then $9 \pi \cdot \frac{50}{360}$, which is $\frac{5}{4} \pi$ ft$^ 2 \approx 3.93 {\text{ ft}}^{2}$ .

To find the area of the isosceles triangle, note that its base is $2 \times 3 \text{ ft} \times \sin \frac{{50}^{\circ}}{2}$, while the height is $3 \text{ ft} \times \cos \frac{{50}^{\circ}}{2}$. Thus, the area is
$\frac{1}{2} \times {\text{base" times "height" = 1/2 times 2 times 3" ft" times sin(50^circ)/2 times 3" ft" times cos (50^circ)/2 = 4.5" ft"^2times sin50^circ ~~ 3.45" ft}}^{2}$

Thus, the area of the shaded part is
$3.93 {\text{ ft"^2- 3.45" ft"^2 = 0.48" ft"^2 ~~ 0.5" ft}}^{2}$(to the nearest tenth)