Possible semi-major and semi-minor axes of an elliptical cylinder based on known surface area and height

41 Views Asked by At

Given a constant surface area $A$ and a known height $h$, how can I manipulate the following equation to isolate for $x$ (or $y$)?

$$A=2\pi xy + h\pi(x+y)(1+\frac{3\frac{(x-y)^2}{(x+y)^2}}{10+\sqrt{4-3\frac{(x-y)^2}{(x+y)^2}}})$$

with

$$ x, y >0$$ (Formula based on Ramanujan's second ellipse circumference approximation.)