Polygon inscribed within an ellipse

To inscribe an equalateral polygon within a circle is easy and it is possible to know (given the size of the circle and the number of sides of the polygon) the exact length of one of the sides of the polygon but what about inscribing an equalateral polygon within an ellipse? To get the length of one of the sides in a circle the formula

R * sqrt (1-cos (360/n) *2) can be used ( among others )

where n= number of sides of polygon

R= Radius of circle

But this does not work for an ellipse

Now remember the polygon must have equal length of sides which must mean that the interior angles at the vertices must be different. I have already worked out the length of side for a twelve sided polygon inscribed within an ellipse of 360 major axis and 240 minor axis as 78.24868802 but have no method other than long winded trial and error to arrive at the result. Can anyone come up with a solution.