Using Area to Prove the Arithmetic-Geometric Mean Inequality

If we have real numbers a and b, we call  \frac{a + b}{2} the arithmetic mean (AM) and \sqrt{ab} the geometric mean (GM) of a and b.  In this post, we are going to examine the relationship of these two means.

To start off, let’s have a few examples.  If a = 3 and b = 12, then GM = 6 and AM = 7.5;  if a = 4 and b=16, then GM = 8 and AM = 10; if a = 3 and b = 27, then GM = 9 and AM = 15. What do you observe? Try a few example and see if your observations hold.

From a few examples above, and from your trials, you have probably observed that GM \leq AM which means that \sqrt{ab} \leq \frac{a + b}{2} for some positive real numbers a and b.  This is actually true for all positive real numbers a and b.  In the following discussion, we are going to use the concept of area to prove that the statement is true.

To begin the proof, we construct a square with side length a + b made up of four rectangles and a square at the center (technically, a square is also a rectangle). Clearly, the area of each of the four rectangles is ab, and the square at the center has are (a - b)^2 (Can you see why?).  If we remove the square at the center, the remaining area is represented by the equation (a + b)^2 - (a - b)^2=4ab. Note that 4ab is the total area of the four rectangles. » Read more