Solution: let the side length of the bottom surface of a regular triangular prism be a, the height be h, and the spherical radius be r,

The height of the triangle at the bottom is (√3)a/2, so there are:

R2=(h/2)2+[(2/3)(√3)a/2)]2, so it catches the ball externally.

Surface area = 4/3 *π, R2 can be obtained.