Two identical spheres of radius R intersect each other such that their centertocenter distance is R. Find the volume enclosed by these two spheres.
If the two spheres were separate their total volume would be (8/3)pi*R^3, but they overlap. The overlapping volume is that of two spherical caps, each with height h=R/2, so one of these caps has volume (pi*R^2/4/3)*(3*R  R/2), twice this is
(pi*R^2/12)*5*R = 5*pi*R^3/12
That is the volume enclosed by both these spheres, that is, the intersection of the two spheres. To get the volume of the union of the two spheres we need to subtract out the volume counted twice in the sum of the separate spheres:
(8/3)*pi*R^3  (5/12)*pi*R^3 = (27/12)*pi*R^3 = (9/4)*pi*R^3

Posted by Charlie
on 20190403 23:18:59 