I wrote two blog posts about this some time ago:

https://www.solipsys.co.uk/new/ConsideringASphere.html?uh14h...

https://www.solipsys.co.uk/new/VolumeOfASphere.html?uh14hn

Pretty sure I posted them here on HN and they sank without trace, but readers might find them interesting.

Simplest way I know to calculate volume of a sphere uses Cavalieri's principle and shows that the volume of an hemisphere is equals to the volume of the difference of a cylinder and a cone. Simple and beautiful.