How does the geometric mean visualize the transformation of a rectangle into a square of equal area?

Conditions

• Given a rectangle with side lengths $a$ and $b$
• Goal is to reshape into a square preserving area

Reasoning, step by step

1. Calculate the area of the rectangle: $A = a \times b$.
2. Find the side length of a square with area $A$: $s = \sqrt{A}$.
3. Recognize that $s = \sqrt{ab}$ is the geometric mean of $a$ and $b$.

Example

A rectangle with sides 9 and 16 has an area of 144. To make a square with area 144, the side length must be $\sqrt{144} = 12$. Thus, 12 is the geometric mean of 9 and 16.

Common misconceptions

• Confusing perimeter preservation with area preservation.
• Thinking the geometric mean is the average of the side lengths added together.

Watch the explanation

Explore next

• Why is the geometric mean called 'geometric'?
• Does this visualization work for more than two dimensions?