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Other than the trivial case of an ellipse with equivalent major and minor axes (a circle) or the degenerate case where one axis is $0$, is there any known ellipse that has rational length major and minor axes and whose perimeter can be represented in closed form as either an algebraic number, or else an algebraic multiple of $\pi$?*

If, by some chance, the answer is yes and such an ellipse is possible I would very much like to know what proportions one such ellipse has.

EDIT: *Or more generally, an algebraic multiple of an algebraic power of $\pi$.

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  • $\begingroup$ Related: mathworld.wolfram.com/EllipticIntegralSingularValue.html $\endgroup$
    – Intelligenti pauca
    Jun 15, 2021 at 16:29
  • $\begingroup$ Simply set the formula for an ellipse arc length equal to the number you want and solve for the axis length. $\endgroup$
    – Тyma Gaidash٠
    Jun 22, 2021 at 17:16
  • $\begingroup$ @TymaGaidash: That won't necessarily yield a rational axis length given an algebraic arc length, which is what the OP wants. $\endgroup$
    – Michael Seifert
    Jun 22, 2021 at 17:17
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    $\begingroup$ @MichaelSeifert the arc length does not have to be rational. I am entirely content with either an algebraic length or an algebraic multiple of pi. $\endgroup$
    – Mark
    Jun 22, 2021 at 17:21
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    $\begingroup$ It's my understanding that factorial isn't really a closed form any more than an integral is. $\endgroup$
    – Mark
    Jun 22, 2021 at 18:11

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