Question

Find the area of the shaded region.

Answer 1

so, if you notice the inscribed regular pentagon, it "emits" 5 radii from the center of the circle, making 3 equal angles.

well, a circle has a total of 360°, so each angle must then be 360°/5, or 72°.

if we notice the line on the right-side, the diameter of the circle is 16 units, meaning its radius is 8 units.

so a shaded section is really just the area of a segment of a circle with an angle of 72° and a radius of 8, keeping in mind we have 3 segments there.

$\bf \textit{area of a segment of a circle}\\\\ A=\cfrac{r^2}{2}\left[ \cfrac{\pi \theta }{180}~~-~~sin(\theta ) \right]~~ \begin{cases} r=radius\\ \theta =angle~in\\ \qquad degrees\\ ------\\ r=8\\ \theta =72 \end{cases} \\\\\\ A= \cfrac{8^2}{2}\left[ \cfrac{\pi(72) }{180}~~-~~sin(72^o ) \right]\implies A=32\left[ \cfrac{9\pi }{10}~~-~~ \stackrel{\approx}{0.95}\right] \\\\\\ A\approx 9.7785774445044391446\qquad \qquad \stackrel{3~segments}{3A\approx 29.3357323335133174}$