so.. we know the arc is 120 inches and has a chord of 8√3 inches
so... what's the central angle? well
[tex]\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\sqrt{3}}=\theta
\\\\\\
496.2\approx \theta[/tex]
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
[tex]\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\sqrt{3}\\
\theta=136.2
\end{cases}\\\\
-----------------------------\\\\
A=\cfrac{(8\sqrt{3})^2}{2}\left( \cfrac{\pi \cdot 136.2}{180}\ - \ sin(136.2^o) \right)\implies A\approx 161.8\ in^2[/tex]
