Calculating the surface area of sphere above a plane How do I calculate the surface area of the unit sphere above the plane $z=\frac12$?
EDIT: I have been attempting things and I am thinking about parameterizing this... While I know that surface area is given by the double integral of the cross products of partial derivatives of the new parameters, I don't know what to set them to.. (sorry I'm not good with the fancy notation)

The circumference of an infinitesimal ring of the unit sphere between $z$ and $z+\mathrm dz$ is $2\pi\sqrt{1-z^2}$, and its width is $\mathrm dz/\sqrt{1-z^2}$. Thus its surface area is $2\pi\,\mathrm dz$. That is, the surface area of a slab of the unit sphere between two $z$ coordinates (or in fact between any two parallel planes) is simply $2\pi$ times the difference of the $z$ coordinates (or, generally, the distance between the two planes). Thus the surface area of the slab of the unit sphere between $z=1/2$ and $z=1$ is $2\pi\cdot(1-1/2)=\pi$.
