Centroid of area drawn over a hemisphere We were taught an theorem for finding the centroid of an arbitrary area having uniform mass density drawn over Hemisphere. ![enter image description here]( The theorem states that the centroid of and arbitrary area $A$ of uniform mass density drawn over Hemisphere is located at $\frac{B}{A}\cdot R$ distance from base of Hemisphere: $$Y_{\text{centroid}} = \dfrac{B}{A}\times R$$ where * $A$ is total area drawn over Sphere * $B$ is the parallel projection of area $A$ over Base of Hemisphere * $R$ is radius of Hemisphere. I have verified this theorem for some symmetrical shapes drawn on Hemisphere. My questions are: 1. Is there a **name** for this equation/theorem? 2. Is there a simple **proof** for this?

Area $A$ and its projection $B$ are given, in spherical coordinates, by: $$ A=\int_\Omega R^2\sin\theta\, d\theta d\phi, \quad B=\int_\Omega R^2\sin\theta\cos\theta\, d\theta d\phi, $$ where I took the base of the hemisphere in the $x-y$ plane, $\theta$ is the polar angle and $\Omega$ is the integration domain.

The height of the centroid is given then by its $z$ coordinates, so by definition: $$ z_{centroid}= {\int_\Omega R^2 z\sin\theta\, d\theta d\phi \over \int_\Omega R^2\sin\theta\, d\theta d\phi}= {\int_\Omega R^2 R\cos\theta\sin\theta\, d\theta d\phi \over \int_\Omega R^2\sin\theta\, d\theta d\phi}= {RB\over A}. $$