Can $\pi$ and the $\pi$ in radians simplify?

Question

I saw in a proof for the limit $$\lim_{x\rightarrow 0}\frac{\sin(x)}{x}=1$$ that, in one of the steps, you had to take the area of a section of a circle, in which you had to do $\frac{\pi r^2 x}{2\pi}$, $x$ being the angle of the section. Then, they simplified the $\pi$ in the numerator with the $\pi$ in the denominator. Do these $\pi$ mean the same thing? Is this simplification correct?

Answer 1

Absolutely. That's the idea behind radians: $2\pi$ radians goes around a circle, and on a unit circle it moves a distance of $2\pi$ in the process.