## What Does This Mean?

When you have a given amount of fencing, why does the circle always produce the most amount of area. Furthermore, even though we discovered that the square has the highest product in the rectangle vs. square comparison post, why does the square have less area than the circle? To answer these questions, let’s look at the ratio of the area to the perimeter of a square and circle with the same side length and diameter respectively.

As you can see, when the circle and the square share the same diameter and length (respectively), the ratio of the area to the perimeter is in fact the same. More importantly, this shared ratio of 2.5 or 1/2r amongst the square and circle comes from geometric logic. Now, let’s explore how:

Before we continue any further, let’s understand that the above derivations work because we know that the side length of the square of is equivalent to 2r. That is how our perimeter for the square was 8r (2r x 4) and the area was 4r^2 (2r x 2r). However, this only applies when the circle and the square have the same diameter/length. Then, what happens when the square is enclosed inside of the circle?

Here, the circle ends up having a better ratio than the square. More importantly, to truly understand how this works, we will first have to explore another scenario –> where the square and circle share the same perimeter. By doing so, we can see exactly how the circle covers more area and therefore makes the best shape for optimization. For the test, let’s assume that the amount of fencing we have is 40.