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.
Alas, this is where people are getting the ‘best for optimization’ claim from. Despite both the shapes taking up the same amount of fencing / perimeter, the circle covers more area. Now, to find out more about how this works, let’s look at the various formulas:
It is evident that the square does not have any use of π, but the circle has π in both of its formulas. Could this impact the area in any way? Yes. To make this even more easier to understand, let’s examine the perimeter scenario in more depth. To find the diameter of the circle, we performed 40/π, whereas to find the side length of the square, we performed 40/4 as there are 4 sides of equal length. That is what made the difference in the area as the square ends up getting a smaller side length and therefore a smaller area. So, what does this tell us? Well, if the value of pi was actually greater than 4, then the circle would no longer hold the best optimization of area title. The reason being, for a perimeter of 40, the circle (with a pi value of 4.5) would have an area off (4.44^2 x 4.5) 88.71. Meaning, the ratio of the area to perimeter would be: (88.71 / 40) 2.2. Alas, the reason the circle is the best for optimization is the fact that the formula has pi. In particular, when you divide to find the diameter length, you divide by 3.14 and not 4 (for the square). Overall, when you analyze the formulas, you start to understand why various shapes have certain properties. Furthermore, we now know that the reason for the circle’s great optimization is that the value of pi is less than 4(which is all it needs to compare to as the square is one of the other great shapes for optimization). Specifically, when you have a given perimeter and you want to find the area, the value of the diameter is greater than the side length of the square as it is being divided by a smaller value: it is just like saying: is x/3.14 or x/4 greater?
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