Some thoughts on circumference

This is the formula for the circumference of a circle: \[C = 2\pi r\]

It’s very simple. My recollection of how it was taught is as a mystical relationship between \(\pi\) and the circumference, as if it were some magical truth that \(\pi\), of all numbers, would be the number that would satisfy the need to calculate the circumference of a circle.

I’ve come to feel that that’s backwards. At some point in the past, engineers and mathematicians wanted to know the relationship between a circle’s radius (or diameter) and its circumference. There were two questions to be answered:

  1. Given a circle of a specific radius (\(1\), for instance), what is its circumference?
  2. As the radius increases, how does the circumference change?

It took a long time to definitively answer the first question, although approximations were easy enough to come by: A circle of radius \(1\) has a circumference just over \(6\). It was eventually convenient to refer to this “just over” value as \(2\pi\) (for historical reasons, we chose to use a sign to represent what is actually¬†half the circumference of a unit circle; discussions like these would be simpler if we’d decided to use a symbol for the full circumference, such as \(\tau\)… and there is a modern movement to do exactly that, called Tauism).

Eventually, mathematicians found ways to calculate \(\pi\), and then later discovered that \(\pi\) shows up in all sorts of awesome, amazing, fascinating places. However, one place that’s not particularly fascinating is its residence in the calculation of the circumference, because that’s where it came from in the first place. The ancients knew there had to be some multiplier that converted the unit circle radius to its circumference, and worked to figure out what that multiplier was. It happens to be \(3.14159…\). The number itself is not a fascinating fact: It’s an inevitability. It had to exist. The fact that it’s irrational¬†is fascinating, but its mere existence is not.

Another detail of the circumference formula that’s easy to miss while being fascinated with the properties of \(\pi\) (or \(\tau\), if you’re a rabble-rouser) is that the formula for the circumference answers the second question in a way that doesn’t seem fully intuitive. However, it does make sense.

Let’s take a square. If we double the size of a square, we double its perimeter. A unit square has a perimeter of \(4\); a square with sides of \(2\) has a perimeter of \(8\). Likewise, if we double the radius of a circle, we double its circumference. If you look at two circles, one twice the size of the other, this might not “seem” right. But it is.

A circle is, on one level, a polygon of infinitely many infinitely small sides. There’s a direct correlation between the diameter of any polygon and its perimeter. Take an octagon and double each of its sides, and its perimeter will double: This makes sense, since you’re doubling each of its sides. Take a duodecagon and do the same, and get the same result.

So the formula for the circumference takes a principle of polygons and generalizes it to include circles as well: Increasing the radius of a circle or any polygon by a scale factor of \(x\) will increase the perimeter (or circumference) by a scale factor of \(x\). To me, this is a more fascinating aspect of the circumference formula specifically.

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