Geometree

An interesting application of the geometric mean of 2 numbers would be calculating the vertical distance to the roof of a sports dome from any point on the playing field, provided the dome is a hemisphere and the playing field lies on the equatorial plane. The distance from any point to the roof is the geometric mean of 2 distances. The 2 distances would be any 2 OPPOSITE distances to the wall. But what's an application for the geometric mean of 3 numbers? (the cube root of their product) Also, how about 4 numbers? (the 4th root of their product)