 Curved area of a spherical cap

For a sphere of radius $R>0$, the spherical cap above $z=R-h$, with $0\le h\le 2R$, 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 $2\pi r\sqrt{1+(dr/dz{)}^2}dz=2\pi Rdz$. 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.