I was asked by a colleague yesterday about how the formula for the arc length of a circle is derived. I wanted to give them a correct answer, so I said I'd get back to them once I'd thought about it properly. Would the following be a correct derivation?
We know that the circumference of a circle is $2\pi r$ (for some fixed radius $r$) and as such we can write down the following relation $$ \frac{arclength}{circumference} = \frac{s}{2\pi r} =t$$ where $0\leq t \leq 1$. Now, we also know that $t=0$ when $\theta =0$ and $t=1$ when $\theta =2\pi$; furthermore, we know that (for a circle) $0\leq\theta\leq 2\pi$ and therefore $\theta =2\pi t\Rightarrow t=\frac{\theta}{2\pi}$. Hence we conclude that $$\frac{s}{2\pi r} =\frac{\theta}{2\pi}\Rightarrow s=r\theta$$ In a similar fashion we can also deduce the area of an arc sector of a circle $$\frac{Area\,\,of\,\,arc-sector}{Area\,\,of\,\,circle} =\frac{A}{\pi r^{2}} =\frac{\theta}{2\pi} \Rightarrow A=\frac{1}{2}r^{2}\theta$$ Where, in both cases, $0\leq\theta\leq 2\pi$ is measured in radians.
Would this be a correct description? Thanks for your time.