Spheroid - Surface Area

Surface Area

An oblate spheroid with c < a has surface area

S_{\rm oblate} = 2\pi a^2\left(1+\frac{1-e^2}{e}\tanh^{-1}e\right) \quad\mbox{where}\quad e^2=1-\frac{c^2}{a^2}.

The oblate spheroid is generated by rotation about the Oz axis of an ellipse with semi-major axis a and semi-minor axis c, therefore e may be identified as the eccentricity. (See ellipse). A derivation of this result may be found at.

A prolate spheroid with c > a has surface area

S_{\rm prolate} = 2\pi a^2\left(1+\frac{c}{ae}\sin^{-1}e\right) \qquad\mbox{where}\qquad e^2=1-\frac{a^2}{c^2}.

The prolate spheroid is generated by rotation about the Oz axis of an ellipse with semi-major axis c and semi-minor axis a, therefore e may again be identified as the eccentricity. (See ellipse). A derivation of this result may be found at

Both of these results may be cast into many other forms using standard mathematical identities and relations between parameters of the ellipse.

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