How to calculate the center of mass of a cut sphere

Question

I did a study in a CAD software (SolidWorks) to get the center of mass of a sphere that is cut at the bottom.

Using trigonometry I was able to obtain the ratio of the diameter cut under the sphere to the initial center of the sphere.

With the code below I make the values ​​valid:

ClearAll["Global`*"]
diametro = {0, 2, 4, 6, 8, 10};
h = N[Solve[5^2 == h^2 + (diametro[[#]]/2)^2, {h}]]&/@Range[Length[diametro]] /. Rule -> Set;
h = Take[h, All, {2}] // Flatten

Based on the equation of the figure below I obtained the values ​​of the centers of mass for each configuration of sphere:

r = 5; H = h + r y = (3 (2 r - H)^2)/(4 (3 r - H)) // N

The question is:

Is there a different way to get the center of mass through solids?

Answer 1

Define solid:

cutBall[r_, h_] := 
 RegionDifference[Ball[{0, 0, 0}, r], 
  HalfSpace[{0, 0, 1}, {0, 0, -h}]]

solids = cutBall[r, #] & /@ h;

GraphicsRow[
 BoundaryDiscretizeRegion[#, ViewPoint -> Front] & /@ solids]

Compute centroid (we only need z-axis value):

RegionCentroid[#][[3]] & /@ solids

{0., 0.00150046, 0.0241226, 0.125, 0.428572, 1.875}