Intuition behind the Geometric Mean

Given a rectangular region of the plane measuring x units by y units, the geometric mean of x and y is the length of as side of the square with equal area to your rectangular region.

These examples are not both geometrical, but they are from real applications often encountered by statisticians, so perhaps they are intuitive.

Geometric mean. Suppose I make measurements $X_i$ that are strongly right-skewed and experience has shown that it is easier to deal with $Y_i = \log X_i.$ (One example is 'lognormal data' which you can read about on Wikipedia.) If I take the arithmetic mean $\bar Y$ of the $Y_i,$ what does that correspond to in terms of the original $X_i$? The answer is the log of their geometric mean: $$ \bar Y = \frac{1}{n}\sum \log X_i = \frac{1}{n} \log \prod X_i = \log \left(\prod X_i\right)^{1/n}.$$

Harmonic mean. (As a bonus, to finish the usual triad of arithmetic, geometric, and harmonic means.) In the US, fuel efficiency of vehicles is usually measured in miles per gallon (MPG). Suppose I drive downhill for a mile at 40 MPG, drive round for a mile on flat roads at 30 MPG, and drive back up the hill at 20 MPG. What is my average MPG? I have used 1/40 of a gallon, 1/30 of a gallon and then 1/20 of a gallon to go three miles. So gallons per mile are about $$(.025 + .033 + .050)/3 = 0.036,$$ and the 'appropriate average' MPG is $$\frac{3}{1/40 + 1/30 + 1/20} = 27.7,$$ which is the harmonic mean of the three separate MPGs. By contrast, in the metric system, fuel efficiency is usually measured by distance per liter, and the arithmetic mean works fine.