Sphere-Sphere Intersection

Let two spheres of radii and be located along the x-axis centered at and , respectively. Not surprisingly, the analysis is very similar to the case of the circle-circle intersection. The equations of the two spheres are

			(1)
			(2)

Combining (1) and (2) gives

(3)

Multiplying through and rearranging give

(4)

Solving for gives

(5)

The intersection of the spheres is therefore a curve lying in a plane parallel to the -plane at a single -coordinate. Plugging this back into (◇) gives

			(6)
			(7)

which is a circle with radius

			(8)
			(9)

The volume of the three-dimensional lens common to the two spheres can be found by adding the two spherical caps. The distances from the spheres' centers to the bases of the caps are

			(10)
			(11)

so the heights of the caps are

			(12)
			(13)

The volume of a spherical cap of height for a sphere of radius is

(14)

Letting and and summing the two caps gives

			(15)
			(16)

This expression gives for as it must. In the special case , the volume simplifies to

(17)

In order for the overlap of two equal spheres to equal half the volume of each individual sphere, the spheres must be separated by a distance

			(18)
			(19)
			(20)

(OEIS A133749) times their radius, where is a polynomial root.

The surface area of the sphere that lies inside the sphere is equal to the great circle of the sphere , provided that (Kern and Blank 1948, p. 97).