The Sneaky Perimeter of the Ellipse

There is no straightforward, precise formula for the perimeter of an ellipse. The below formula relies on an elliptic integral, not expressible in elementary functions. Calculating the perimeter (P) of an ellipse involves then the use of elliptic integrals, represented by the formula shown in the diagram…
a is the semi-major axis,
e is not Euler’s number but the eccentricity of the ellipse,
θ is the parameter ranging from 0 to π/2.

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