To solve this, let's define the problem more clearly:

• You have two spheres, each with radius $r$.
• The center of each sphere lies on the surface of the other sphere.

This arrangement means the distance between the centers of the two spheres is exactly $r$, which is the radius of each sphere.

The region that is common to both spheres is called the intersection volume. To find the volume of this intersection, we use a standard formula derived from geometry of intersecting spheres. When two spheres intersect in such a way that the distance between their centers is equal to their radius, the volume of their intersection is given by:

$V = \frac{16}{3} \pi r^3$

This formula comes from the geometry of the spherical caps formed by the intersection. The key observation is that the distance between the centers $r$ results in a specific type of intersection.

Explanation:

• The two spheres intersect symmetrically, forming a "lens-shaped" volume.
• This volume can be decomposed into two spherical caps, one from each sphere.
• The formula for the volume of this intersection is derived using calculus and the geometry of spherical caps, but it simplifies to the above expression for the case where the distance between the centers equals the radius.

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Would you like a more detailed step-by-step breakdown of the derivation of this formula?

Here are some related questions:

1. How do you find the volume of a spherical cap?
2. How does the distance between the centers of two spheres affect the volume of intersection?
3. What happens to the intersection volume if the distance between the centers is greater than the radius?
4. Can this formula be applied if the spheres have different radii?
5. How can the volume of intersection change if one sphere is inside the other?

Tip: Visualizing the intersection of two spheres as a "lens" shape can help when trying to understand the geometry of the problem.