# Question #e5881

Jan 28, 2016

#### Answer:

See explanation..

#### Explanation:

Remember that $\pi$ is the ratio between the diameter and the circumference of the circle.

If diameter is $1 , c i$$r c u m f e r e n c e = \pi$
If $2 , c i$$r c u m f e r e n c e = 2 \pi$

So,circumference of circle=$\pi d$$, d = \mathrm{di} a m e t e r$

$r = r a \mathrm{di} u s = \frac{d}{2}$
So,$d = 2 r$

So,circumference of circle=$2 r \pi , \mathmr{and} , 2 \pi r$

This is a simple poof that i think is the best

Jan 28, 2016

#### Answer:

here is how you do it friend,

#### Explanation:

for this, we will need closed line integration.
suppose the radius of a circle is r
and after $\mathrm{dt}$ time it makes $d \theta$ angle in the center of the circle having the arc of $\mathrm{dS}$
so,
$\mathrm{dS} = r d \theta$
now,
$S = {\oint}_{0}^{s} \mathrm{dS}$

$= {\oint}_{0}^{2 \pi} r d \theta$

$= r {\oint}_{0}^{2 \pi} d \theta$

$= r {\left[\theta\right]}_{0}^{2 \pi}$

$= r \left[2 \pi - 0\right]$

$= r 2 \pi$

$= 2 \pi r$