Archimedes Volume of a Sphere
In the diagram above, we have a hemisphere with radius r side by side with a cylinder with radius r and height r
The cylinder has had an inverted cone with the same radius and height removed from its interior.
The diagram at the right shows the cross-sections of both figures. To change the height of the plane, use the slider at the bottom of the diagram. (Cross-sections are intersection of the figure with the plane)
Denote A(h) the area of the annulus ("ring") obtained by intersecting the plane at height h with the cylinder with the cone removed.
Denote by S(h) the area of the circle obtained by by intersecting the plane at height h with the hemisphere.
Find formulae for A(h) and S(h) in terms of h and r.