Question

Express answer in exact form. Show all work for full credit.

A segment of a circle has a 120 arc and a chord of 8 squared 3 in. Find the area of the segment.

Answer 1

so.. we know the arc is 120 inches and has a chord of 8√3 inches

so... what's the central angle? well


`\bf \cfrac{180s}{\pi r}=\theta\qquad \begin{cases} \theta\textit{angle in degrees}\\ r=radius\\ s=\textit{arc's length} \end{cases}\implies \cfrac{180\cdot 120}{\pi 8√(3)}=\theta \\\\\\ 496.2\approx \theta`

now, a full circle has 360°, this angle is 496.2° or thereabouts, which gives us no usable segment, however, let's use the coterminal angle on the first two quadrants, check the picture below, so... we'll be using 496.2 - 360 as our angle for the segment then


`\bf \textit{area of a segment of a circle}\\\\ A=\cfrac{r^2}{2}\left( \cfrac{\pi \theta}{180}\ - \ sin(\theta) \right)\qquad \begin{cases} \theta=\textit{angle in degrees}\\ r=radius\\ ----------\\ r=8√(3)\\ \theta=136.2 \end{cases}\\\\ -----------------------------\\\\ A=\cfrac{(8√(3))^2}{2}\left( \cfrac{\pi \cdot 136.2}{180}\ - \ sin(136.2^o) \right)\implies A\approx 161.8\ in^2`

Answer 2

The radius must be 8/sqrt(3)
So area of entire circle is pi * 64/3
So area of sector is 64pi/9
area of segment = 64pi/9 - 16sqrt(3)/3