Hello, miley_22!

Another approach . . .

There is a circle with radius r and center O.

The points A, B, and C are on the circle

and $\displaystyle \theta = \angle AOC$ subtends arc $\displaystyle \widehat{ABC}.$

The area of the sector $\displaystyle \overline{OABC}\:=\:\frac{4}{3}\pi$

. . and the length of the arc $\displaystyle \widehat{ABC} \:=\:\frac{2}{3}\pi$

Find the value of $\displaystyle r$ and $\displaystyle \theta$. Code:

* * *
* * A
* o
* / o B
/
* / θ *
* * - - - - o C
* O *
* *
* *
* *
* * *

The area of the sector is: .$\displaystyle A \:=\:\frac{1}{2}r^2\theta \:=\:\frac{4}{3}\pi$ .**[1]**

The arc length of $\displaystyle \widehat{ABC}$ is: .$\displaystyle s \:=\:r\theta\:=\:\frac{2\pi}{3}$ .**[2]**

Divide [1] by [2]: .$\displaystyle \frac{\frac{1}{2}r^2\theta}{r\theta} \:=\:\frac{\frac{4\pi}{3}}{\frac{2\pi}{3}}\quad\Ri ghtarrow\quad\frac{1}{2}r\:=\:2\quad\Rightarrow\qu ad\boxed{ r \:=\:4}$

Substitute into [2]: .$\displaystyle 4\theta\:=\:\frac{2\pi}{3}\quad\Rightarrow\boxed{\ theta \:=\:\frac{\pi}{6}}$