Spherical Cap Volume

Slice a sphere with a flat plane and the smaller piece is a spherical cap. Its volume, $V = \ frac{\pi h^2}{3}(3R - h)$, depends on the sphere's radius $R$ and the cap's height $h$, measured from the flat cut to the top of the dome. Archimedes solved this by exhaustion in the third century BC, long before calculus made it a routine integral, and he was proud enough of his sphere results to have one carved on his tomb.

The formula earns its living gauging tanks. Pressure vessels and road tankers are built with dished or hemispherical ends, and the liquid sitting in the bottom of a spherical tank is exactly a cap, so this is the equation behind the calibration chart on the side of the vessel. Architects use it for domes, opticians for the sagitta of a lens, and mapmakers for the area and volume of a polar region.

Two boundary cases make good self-checks. At $h = R$ the expression gives $\ frac{2}{3}\pi R^3$, a hemisphere; at $h = 2R$ it gives $\ frac{4}{3}\pi R^3$, the entire sphere. Anything beyond $2R$ is not a cap at all, which is why the solver rejects it. Note also that you can solve this relation for $R$ in one line but not for $h$: isolating the cap height leaves the cubic $h^3 - 3Rh^2 + 3V/\pi = 0$, which has no useful elementary root, so that direction is deliberately not offered here.