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Geometric Mean
Contributed by: Miles
(Original author: Smith)
In this activity, you will learn about the geometric mean.
Let's begin by reviewing a related concept: the "arithmetic mean."
The arithmetic mean is a formal way of saying average 
To find the arithmetic mean of two numbers, we add thenumbers and divide by 2.
What is the arithmetic mean of 4 and 16?
Answer:
The "arithmetic mean" of 4 and 16 is 10, 
since (4 + 16) ÷ 2 = 10

The geometric mean is another "type" of average.
To find the geometric mean of two numbers, you will multiply the two numbers and then take the square root.
What is the geometric mean of 4 and 16? 

Answer: 
4 • 
=
=
The geometric mean of two numbers can be 
calculated by finding the SQUARE ROOT of the 
product of both numbers.
The geometric mean of 4 and 9 is 6.
√
4 * 9
=
The geometric mean of two numbers can be 
calculated by finding the SQUARE ROOT of the 
product of both numbers
The geometric mean of 2 and 8:
√
2 * 8
=
The geometric mean of two numbers can be 
calculated by finding the SQUARE ROOT of the 
product of both numbers
The geometric mean of 4 and 25:
√
4 * 25
=
The geometric mean of two numbers can be 
calculated by finding the SQUARE ROOT of the 
product of both numbers
The geometric mean of 2 and 32:
√
2 * 32
=
The geometric mean of two numbers can be 
calculated by finding the SQUARE ROOT of the 
product of both numbers
The geometric mean of 3 and 12:
√
3 * 12
=
We can try to understand geometric mean by using an area model.
On the previous slide, we solved that the geometric mean of 12 and 3 is 6.
This can be interpreted geometrically this way: 
The area of a rectangle with dimensions 12 and 3is equivalent to the area of a square of side length 6.
12 
3
6
6
The geometric mean of two numbers can be 
calculated by finding the SQUARE ROOT of the 
product of both numbers
The geometric mean of 2 and 50:
√
2 * 
=
The geometric mean of two numbers can be 
calculated by finding the SQUARE ROOT of the 
product of both numbers
The geometric mean of 3 and 27:
√
3 *
=
The geometric mean of two numbers can be calculated by finding the SQUARE ROOT of the product of both numbers.
The geometric mean of 5 and 7:
√
5 * 7
=
√
Note: The GEOMETRIC MEAN does NOT have to be
a whole number, or a rational number
(it can be an irrational number).
The geometric mean of 5 and 7:
√
√
35
5*
7
is the geometric mean.  
It cannot be simplified.
=
√
35
Find the geometric mean of 5 and 11:
√
5 *
fill in both
blanks
=
√
When we drop an altitude from the right angle of a righttriangle to the hypotenuse, three similar right triangles are formed. 
When we drop an altitude from the right angle of a righttriangle to the hypotenuse, three similar right triangles are formed. 
The two smaller right triangles will be similar to each other and also similar to the large, outer right triangle.
We can write a ratio of height : base and solve for "x".
height
base
x2 = 9 • 16
x = √ 9 • 16
x = 
x
9
=
16
True or False: The length of the altitude "x" is the
geometric mean of the two segments of the hypotenuse.
Look again at the right triangle diagram, with altitude marked as "x":
True
False
x2 = 9 • 16
x = √ 9 • 16
x = 12
The length of the altitude "x" is the geometric mean of the two segments of the hypotenuse.  Solve for "x".
x = √
x
?
  2 = 36 • 
x = 
x = 
6 • 8
?
48
?
36 • 64
?
64
The length of the altitude "x" is the geometric mean of the two segments of the hypotenuse.  Solve for "x".
2304
48
?
2304  36
?
  2 = 36 • 
x = 
=
64
?
= x 
36x
?
x
?
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