In addition to the arithmetic mean and the harmonic mean, we also have a way of "averaging" that is called the geometric mean. Together, the geometric mean, the arithmetic mean, and the harmonic mean are referred to as the Pythagorean Means. 

The geometric mean of n numbers is written as follows:

G =

In other words, multiply all the values together, and then take the n^th root.

Example
What is the geometric mean of 8 and 32?

Solution
Multiply the two numbers together: 8 x 32 = 256. Now take the square root: = 16.

As with the harmonic mean, this is skewed toward the lower end. The arithmetic mean of 8 and 32 is bigger than the geometric mean: = 20.

Example
What is the geometric mean of 2, 3, and 36?

Solution
Multiply the three numbers together: 2 x 3 x 36 = 216. Now take the cube root (because we have three numbers): = 6.

When to Use
The geometric mean shows up (not surprisingly) in geometry. If you draw the altitude (A) to the hypotenuse in a right triangle, the altitude splits the hypotenuse into two pieces (B and C) and splits the triangle into two similar triangles. Because of proportional sides in the similar triangles, we have:

=

This results in A² = BC or A = .

In other words, the length of the altitude is the geometric mean of the two segments of the hypotenuse.

1.

What is the geometric mean of 2 and 8?

2.

What is the geometric mean of 4, 4, and 108?

3.

The geometric mean of two numbers is 12. One of the numbers is 9. What is the other number?

4.

The geometric mean of three numbers is 6. Two of the numbers are 3 and 72. What is the other number?

5.

The hypotenuse of a right triangle is split into two pieces by the altitude. The lengths of the two pieces are 4 and 6. What is the length of the altitude?

6.

The hypotenuse of a right triangle is split into two pieces by the altitude. The length of one piece of the hypotenuse is 4 times the length of the other. The length of the altitude is 18. What is the length of the hypotenuse?

7.

The arithmetic mean of two numbers is 4.5,and their harmonic mean is 4. What is their geometric mean?

8.

The geometric mean of 1 and X is an integer. What can you state about X?