I know that the area of the sector of circle can be computed using the following two formulas
$$\pi r^2 \frac{\theta }{360} \space \space \text{ (degrees case)}$$ $$or$$ $$r^2 \frac{\theta }{2} \space \space \text{ (radians case)}$$
The first case makes complete sense. You just consider the fraction of the full 360 degree circle and use this fraction to scale the full area formula.
The second case doesn't make much sense. I've searched for proofs and most relied on calculus, which seems like overkill... Is there a simpler way of showing that the radian case is valid?
As an attempt I tried converting the degrees to radians using $\frac{\pi}{180}$
$\pi r^2\frac{\theta \space \cdot \frac{\pi}{180}}{360}= \pi^2 r^2 \frac{\theta}{64800} \neq r^2 \frac{\theta}{2}$
Question: Why is my approach for converting the degree case of the formula to the radian case of the formula incorrect?