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)

 

One application would be in computing compound annual growth rates.

Geometric means are important in evolutionary biology. Evolution is a multiplicative process when looked at over several generations.

This is why, according to bet hedging theory, natural selection should favor strategies that maximize the geometric mean over long timescales.

 

This is not necessarily always realized. For example if the environment is stable for a very long time, and then there is one very very bad year, strategies that are better over short time-scales might invade the population during the stable period. Nevertheless, it is an important principle in evolutionary biology.