What is the simplest formula for calculating the circumference of a circle?

Question

I am studying for a test to become a "paraeducator/instructional aid." One of the pages in the study guide has this information on computing area and perimeter and circumference and such:

Wouldn't it be simpler to say "multiply the diameter by pi"?

Answer 1

Conservation of Difficulty

You should not be focused on what formula is "simpler". I have often joked with colleagues that mathematics has a kind of Law of Conservation of Difficulty: any particular problem has a certain difficulty associated to it, and there is not really any way of getting around that difficulty. My usual go-to example of this is a proof of the Fundamental Theorem of Algebra: there are very elementary proofs which basically boil down to gradient descent, and fancier proofs which rely on Liouville's theorem.

The elementary proofs are typically very long and tedious—because an elementary proof uses elementary tools, there are no big theorems which can be used to make the proof shorter or "simpler"—one must simply brute force the problem. On the other hand, a proof using Liouville is a two liner—but one has to spend a lot of time building up the theory of complex analysis before getting to a proof of Liouville so that one can then proceed to prove the Fundamental Theorem of Algebra. The difficulty of the proof can be moved to a different place, but it can't be entirely removed.

As an instructor, it is your job to take complicated ideas and find a path through those ideas which will develop the theory in a coherent manner, so that the steps from one idea to the next will each appear simple, until you have a chance to zoom out or look back and see the material taught in total.

An Ahistorical History

In geometry, a circle is defined by two things: a point $C$ (the center of the circle) and a distance $r$ (the radius of the circle). By definition, the circle with center $C$ and radius $r$ is the set of points which are exactly $r$ units from $C$. In classical Euclidean geometry, this makes sense, since one of the basic tools is the compass, which can be used to construct a circle in exactly the manner described above, the center and radius are fundamental.

This implies that any formula relating to a circle should start from these two data: the center and the radius. After some investigation, we learn that the circumference of a circle is proportional to the radius. That is, there exists some constant $k$ such that the circumference $P$ of a circle with radius $r$ is $$ P = kr. $$ A bit more work shows that $k \approx 6.28$. This number shows up in a lot of places, so one might want to go ahead and give it a name: $\tau ≈ 6.28$ is, by definition, the ratio of the circumference of a circle to its radius.

However, in the real world, measuring the radius of a circle is a bit of a pain—one has to find the center of the circle, and then measure the radius. It turns out that it is easier to measure the diameter (e.g. with calipers; or by constructing a chord and considering a perpendicular bisector of that chord). So, from a practical point of view, it is often easier to get one's hands on the diameter, $D$, of a circle. The diameter is twice the radius, so another formula for the circumference of a circle is $$ P = \tau \frac{D}{2} = \frac{\tau}{2} D. $$ Because the term $\tau/2$ appears repeatedly, it might be wise to give this a name, too. So define $$ \pi = \frac{\tau}{2} = \frac{\text{circumference}}{\text{diameter}} \approx 3.14. $$ At the end of the day, there are two (equivalent) formulae for the circumference of a circle: $$ P = \pi D \qquad\text{and}\qquad P = 2\pi r. $$ The former is, perhaps, more practical (because it is, maybe, easier to measure a diameter); but the latter is more fundamental, as it relates directly to the construction of the circle in terms of a center and radius.

The Language of Mathematics

Another job of an instructor is to teach students the language used in their field. In mathematics,

• one "multiplies two numbers" (one does not "times two numbers);
• one "evaluates an integral" (one does not "solve an integral);
• one "solves a polynomial equation" and "determines [or finds] the zeros of a polynomial function" (one does not "solve a function");
• and so on.

The use of the radius in the formula for the circumference of a circle is a matter of correct mathematical "spelling" or "grammar"—it is the correct way to use language as mathematicians use that language. Whether or not it is "more simple" (which, as I have argued above, it really isn't), it is how the language is used, and students need to get used to that use of language.

This is particularly important, as there are many, many other places in mathematics where the radius plays a role. In three-dimensional geometry, it shows up in the formulae for the volume and surface area of a sphere; it is fundamental to the definitions of the cosine and sine functions as functions on $\mathbb{R}$; and so on. The radius shows up everywhere in the language of mathematics, so one might as well get students used to this as early as possible.

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Tangential Note: In the formula sheet quoted in the question, it is asserted that $\pi = 3.14$. This is wrong—don't say this to students; $\pi$ is not equal to $3.14$. This is a reasonable approximation for $\pi$ in most circumstances, but there is no equality here. The constant $\pi$ is irrational—it does not have a terminating or repeating decimal expansion. It is totally reasonable to tell students that $\pi \approx 3.14$, but be clear that this is an approximation, not an equality.

Answer 2

You could say: $\tau r$.

Circles are defined geometrically by their radius (note it has precedence before diameter in the study guide), so a lot of formulas would be conceptually more meaningful, and often simpler, if the radius was in the picture. On the other hand, real-world circles are arguably easier to measure via the rim-to-rim diameter. As a result, we've gotten in the historically unfortunate habit of shifting a factor of $2$ from the circle constant to the circle measure.

A lot of people now note that many formulas would be simplified if the circle constant used was $\tau = 2 \pi$. E.g., that's the number of radians in one full circle (whereas $\pi$ only gets you halfway around). And then like many other formulas, here you're looking at a factor of $r$, which is more fundamental than the diameter in the definition of a circle (e.g., as in the area formula immediately prior in the given study guide).

This larger story is probably why the study guide prefers to see $r$ in the formula. It's part of other circle-based formulas (as opposed to diameter), so students should be most familiar with it.

See more at: The Tau Manifesto.

Answer 3

It is a matter of taste. But there are some reasons for this preference.

1. It is easiest to use either the radius or the diameter and to stick with it. There is less conversion, so less mental load and less room for error. So given that we absolutely want this to be the case and we also only consider the formulas for area and circumference, we would have to choose between $A=\pi r^2$ and $C=2\pi r$ or $A=\tfrac 1 4\pi D^2$ and $C=\pi D$. Just from these formulas I would not say there is a clear winner.
2. How about other dimensions? We can do the same comparison for higher dimensions and compare which expressions look nicer.

The formulas using $r$ behave better when the number of dimensions get larger. When using the diameter in 6D, one of the coefficients is 384, yuck!

3. When you start doing calculus, at some point you will likely calculate the area of a circle using integration. For that you will use some form of the formula $x^2+y^2\leq r^2$. Not only is the radius a clear winner here in terms of computation, conceptually it is also easier to define it in terms of radius. The formula $x^2+y^2\leq r^2$ is also used a lot in programming.

So to summarise, the decision is kind of arbitrary. But some considerations that come from more "advanced" uses, combined with a fair bit of historical bagage, manage to tip the scale towards the radius in favor of the diameter. There are some things to be said though for the diameter. First of all, $\pi$ is often defined as the ratio between the circumference of the circle and its diameter, i.e. $\pi=C/D$. Also, in some areas in physics (for example soft matter) the size of spheres is defined using the diameter instead of the radius.

For those interested, I used the following Mathematica code to create the tables:

TableForm[
 Table[{ToString[d] <> "D", (2*Pi^(d/2)*r^(-1 + d))/
    Gamma[d/2], (Pi^(d/2)*r^d)/Gamma[1 + d/2]}, {d, 1, 10}], 
 TableHeadings -> {None, {"Dim.", "Area", "Volume"}}]

Answer 4

Apologies for the commenty-answer, but have not seen this aspect raised yet:

In no way, wanting to cut you off from wondering about math or the like. But you should concentrate on teaching the standard methods, content, notation, etc. Especially as an "aide". But even if you were a head of the NYC public schools or the like.

There are plenty of more important issues like how to teach fractions or keep control in the classroom. (And these are non-trivial!)

Of course, every question can't (and shouldn't) be on the highest priority topic. But I just want to warn you.

Answer 5

Given an arc-length of $l$ which represents a proportion, $p$ of the circle, the circumference is $\frac{l}{p}$. For example, if an arc-length is $1$ and that represents $\frac{1}{3}$ of the circle, then the circumference is 3.

Some may scoff at division being "simpler," but with just 2 terms, no constants to remember, and no $\pi :\tau$ debates, $\frac{l}{p}$ is hard to ignore.

Answer 6

Well, the fun fact is, diameter does not "exist" in math. All formulas are defined in terms of radius r.
The only time you will see diameter d mentioned is in the formula introducing concept of Pi (ratio of circumference to diameter)

Another fun fact is that you will see that "2 Pi" pop up in majority of formulas. Truly, it would be more convenient to have a constant of value 2 Pi (*).

(*)Which we actually have, look for Tau Manifesto.