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Hy-Spherical (Posted on 2019-04-03) Difficulty: 3 of 5
Two identical spheres of radius R intersect each other such that their center-to-center distance is R. Find the volume enclosed by these two spheres.

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution if my algebra was careful enough... Comment 1 of 1
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 2019-04-03 23:18:59
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