Equation to calculate the cap area of an oblate spheroid
I found an equation in this publication that seems to produce correct results:
$$ e = \sqrt{\frac{a^2}{c^2}-1}, $$ $$ e_{1} = e\left(1-\frac{h}{c} \right) $$ $$ A = \pi a c \left(\frac{\sinh^{-1}e - \sinh^{-1}e_1}{e} + \frac{a}{c} - \left( 1-\frac{h}{c}\right)\sqrt{1+e_{1}^{2}} \right) $$