= Curved area of a spherical cap
{title2=$A=2\pi Rh$}
For a <sphere> of <radius> $R>0$, the <spherical cap> above $z=R-h$, with $0\leq h\leq2R$, has curved <surface area> $2\pi Rh$. On the surface use $r(z)=\sqrt{R^2-z^2}$. The <surface of revolution> area element is $2\pi r\sqrt{1+(dr/dz)^2}\,dz=2\pi R\,dz$. Integrating from $R-h$ to $R$ proves the formula. It includes only the spherical surface; if the planar closing disk is required, add $\pi(2Rh-h^2)$. The coordinate expression is interpreted by a <limit> at a pole.
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