Centroid of area drawn over a hemisphere

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We were taught an theorem for finding the centroid of an arbitrary area having uniform mass density drawn over Hemisphere.

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?

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1 Answer

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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: $$ z_{centroid}= {\int_\Omega R^2 z\sin\theta\, d\theta d\phi \over \int_\Omega R^2\sin\theta\, d\theta d\phi}= {\int_\Omega R^2 R\cos\theta\sin\theta\, d\theta d\phi \over \int_\Omega R^2\sin\theta\, d\theta d\phi}= {RB\over A}. $$

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