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Quirky Question which is thought provoking:

Why isn't there an exact formula for the circumference of an ellipse?

John Douma
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Ayush Sambher
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    There are a lot of exact formulas, but there is not an exact formula for certain restrictions on what can be used to build/express the formula. For example, there is an exact formula for the positive solution to $x^2 = 2,$ namely $\sqrt {2},$ but there is not an exact formula for the positive solution to $x^2 = 2$ if the formula is restricted to be a an expression involving a finite number of integers and a finite number of the operations of addition, subtraction, multiplication, and division. – Dave L. Renfro Jun 28 '21 at 10:55
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    What makes you think there isn't an exact formula? Would you agree $2\pi r$ is the exact formula for the circunference of a circle? Keep in mind that $\pi$ has an infinite series representations much like the solution for the circumstance of an ellipse. Neither can be calculated with a finite number of steps. – Aaron Hendrickson Jun 28 '21 at 11:38
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    If we take a sample of "simple" closed curves, the majority of curves will not have an "exact" formula for their circumference. – MasB Jun 28 '21 at 11:41

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