[tex]\bf \textit{area of a sector of a circle}\\\\
A=\cfrac{\theta \pi r^2}{360}\quad
\begin{cases}
r=radius\\
\theta =angle~in\\
\qquad degrees\\
------\\
A=8\\
r=4
\end{cases}\implies 8=\cfrac{\theta \pi 4^2}{360}\implies \cfrac{8\cdot 360}{4^2\pi }=\theta
\\\\\\
\cfrac{2880}{16\pi }=\theta \implies \boxed{\cfrac{180}{\pi }=\theta }\\\\
-------------------------------\\\\[/tex]
[tex]\bf \textit{arc's length}\\\\
s=\cfrac{\theta \pi r}{180}\quad
\begin{cases}
r=radius\\
\theta =angle~in\\
\qquad degrees\\
------\\
\theta =\frac{180}{\pi }\\
r=4
\end{cases}\implies s=\cfrac{\frac{180}{\underline{\pi} }\underline{\pi} \cdot 4}{180}\implies s=\cfrac{\underline{180}\cdot 4}{\underline{180}}
\\\\\\
\boxed{s=4}[/tex]
if you do a quick calculation on what that angle is, you'll notice that it is exactly 1 radian, and an angle of 1 radian, has an arc that is the same length as its radius.
that's pretty much what one-radian stands for, an angle, whose arc is the same length as its radius.
