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**Prove It** A radian is a length of the radius on the circumference.

You're told the radius is $\displaystyle r$ and that the length of the arc $\displaystyle PQ$ is $\displaystyle 6\textrm{bf}$.

Therefore the angle $\displaystyle \theta$, in radians, is $\displaystyle \frac{6}{r}$.

The area of a sector, if the angle is given in radians, is

$\displaystyle \frac{\theta}{2\pi}\pi r^2 = \frac{\theta}{2}r^2$.

We know that $\displaystyle \theta = \frac{6}{r}$ and that the area is $\displaystyle 22.5\textrm{cm}^2$.

So $\displaystyle \frac{\frac{6}{r}}{2}r^2 = 22.5$

$\displaystyle \frac{3}{r}r^2 = 22.5$

$\displaystyle 3r = 22.5$

$\displaystyle r = 67.5\textrm{cm}$. correction, r = 7.5 cm

From this, we can see $\displaystyle \theta = \frac{6}{67.5}^C$.