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

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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:

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

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

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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

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