# How do you prove that the circumference of a circle is #2pir#?

##### 3 Answers

Basically this is a definition thing.

This ratio is a constant since all circles are geometrically similar and linear proportions between any similar geometric figures are constant.

If you were looking for how the value of the ratio

If we imagine the circle centered in the origin with radius

graph{x^2+y^2=4 [-10, 10, -5, 5]}

or

And considering the fourth of circle in the first quadrant, we can obtain the lenght of a line with the integral:

This integral is quite long, so we can parametrize the circle as usual:

and use this integral:

Since:

So:

I don't think you can prove it, because that is, or is equivalent, to the definition of

#### Explanation:

Similarly I don't think you prove that *Evaluating*