# How do can you derive the equation for a circle's circumference using integration?

Let assume that the circle is centered at the origin hence its
equation is

${x}^{2} + {y}^{2} = {r}^{2}$

Using parametrization in two variables: we can write the same circle above as

$\left(x \left(t\right) , y \left(t\right)\right)$,with $x \left(t\right) = r \cos t$,$y = r \sin t$,0≤t≤2π

and thus the arclength is given by (integration)

$\setminus {\int}_{0}^{2 \setminus \pi} \setminus \sqrt{x ' {\left(t\right)}^{2} + y ' {\left(t\right)}^{2}} \mathrm{dt} = \setminus {\int}_{0}^{2 \setminus \pi} \setminus \sqrt{{r}^{2} \setminus {\sin}^{2} t + {r}^{2} \setminus {\cos}^{2} t} \mathrm{dt} = r \setminus {\int}_{0}^{2 \setminus \pi} \mathrm{dt} = 2 \setminus \pi r$