 ## The Way of the Tau (why Tau is better than Pi)

If you’re a math geek, you’ll find that Tau is a lot tastier than Pi!

Originally published 2015

Tau (τ) is a mathematical constant equal to two times Pi (π). You probably learned about π in school. But as you’ll see, you should have been learning about τ — if only because it makes things a world easier to understand.

If you know me, you know that I am all about usability. If there is a proposal that makes things more comprehensible, easier to understand, or generally more usable, then I am going to be in favor of that proposal. In part, that’s because of the multiplier effect. A little work making things easier to use is multiplied by hundreds and thousands of uses. So the initial effort is repaid many times over. In part, it’s simple human decency. Why make things any harder to use or to understand than they need to be?

The idea of replacing Pi with Tau is one of those proposals. It is, quite simply, the right thing to do — and it is going to take very little time to show you why.

The value of Pi is approximately 3.14 — or more accurately, 3.1419…, with digits that continue to infinity without ever repeating. So “Pi day” is March 14 (3/14). Tau is 2 times Pi, or approximately 6.28 — or more accurately, 6.2832… So “Tau day” is 6/28.

So some of the more amusing reasons for preferring τ to π are:

• You can have two pie on Tau day. rather than just one on Pi day.
• If you start celebrating π on 3/1, and continue until 3/14, you have two weeks.
But if you start celebrating τ on 6/2 and continue until 6/28, you have very nearly the whole month!

Seriously, though, there are much more important reasons for preferring τ. If you consult the references, you’ll find nearly a dozen formulas that depend on the constant “2π“. In fact, almost every formula in math, engineering, and physics that uses π at all uses the constant, 2π.

As Joseph Lindenberg writes in Tau Before It Was Cool:

The idea that 2π, not π, is the special number that deserves its own symbol, first occurred to me in the fall of 1988, when I was an undergraduate student in electrical engineering.  After having seen 2’s next to π’s for years in countless equations, the reason for all those hangers-on 2’s finally sank in one evening. Mathematics was using the wrong constant!

But others have made the same observation independently. Bob Palais’ paper Pi is Wrong kicked off a lot of academic debate on the subject. But the person who has done the most to bring it to people’s attention is undoubtedly Michael Hartl, who wrote the The Tau Manifesto. (He even has T-shirts! You can wear them on Tau day to help spread the gospel.)

But again, the bottom line for me is usability — or in this case, ease of understanding. Let’s see what τ does for us, on that score.

Contents

## For Circumferences, Tau Makes Sense

Let’s take a very practical problem. You have a small garden, and part of it has a circular border. How can you tell much fencing do you need to cover that arc? As you’re going to see, knowledge of Tau is going to make that problem pretty darn easy to solve.

To make things as simple as possible, let’s say you have a wedge-shaped garden that looks like a piece of pie: You can step off one side of the wedge to get the radius. And you know the wedge is 1/4 of the circle. Do you know how to calculate the circumference of that wedge?

Think about it for a second before moving on. I know you only have to think for a second, because unless you studied this particular formula just last week, there is virtually no chance at all that you will remember the calculation. So let me ask you: Did you remember that formula? I thought not! How could you, unless you use it every day. But in a second I am going to show a way of thinking about the problem that is so simple, you will never, ever forget it again.

We’ll start by understanding exactly what τ is:

If the radius of the circle is one unit, then the circumference is τ units (6.28…).

In a nutshell, you can think of the constant τ as something that stands for “one turn” of the wheel.
If the radius of the circle is 1, then circumference is Tau! (τ`*r`, where` r=1`). There are other ways you can use the “one turn” concept. For example: If a 26″ bicycle wheel has a radius of 13 inches, how far will the wheel travel in one revolution? Simple: 13 * τ, or 13 * 6.28, (off to the calculator…) voila! 81.64 inches!

Now, then. You have a wedge that is 1/4 of circle. What is the length of the circumference on that wedge? Again, think it about it for a second. And again, it should it only take a second!

If the full circumference of the circle is τ*r, then 1/4 of the circle has a circumference equal to 1/4 *τ*r! Simple, right? Unlike π, τ has a meaning that is readily comprehensible and immediately applicable.

Now that you know what τ really is, you’re going to find that the concept of radians makes a lot more sense, as well.

A radian is an angle that produces an arc that is equal to the radius: In mathematics, radians turn out to be helpful in a variety of ways. It’s beyond the scope of this article to describe the many contexts in which radian measurements are useful (and beyond my current math skills, actually). But its pretty easy to understand what they are. And more to the point, you can now determine exactly how many radians there are in a circle.

So, how many radians are there in a circle? If you don’t recall that there are “2π” radians well, I for one do not blame you. Who would remember that? How would they remember it — unless they happened to use it all the time.

But if a radian carves out an arc of length equal to the radius (“1”, in this case), and if a circle with radius 1 (the unit circle) has a circumference of length τ (6.2832…), now how many radians are there in a circle? Yup. It’s pretty simple, huh? There are precisely τ radians in a circle. And that’s true for any circle, regardless of its size.

Makes sense, huh? It’s pretty easy to understand, given the concept of τ. And if that was the constant you learned in school, all of this would be second nature, even if you hadn’t studied a lick since. (Who of us has?)

## What About the Area of a Circle?

The formula for the area of a circle is so pleasant: π`r2`. Using τ, the formula becomes `(`τ`r2``)`/2. That’s more of a headache. So isn’t it worth keeping π for that formula, if nothing else?

Ok. In all honesty, it probably is. But as Michael Hartl points out in The Tau Manifesto, there are many other examples in which that factor of 1/2 shows up:

• The distance an object falls in a unit of time: `(gt2)/2`
• The amount of potential energy stored in a spring: `(``kx2)/2`
• The amount of kinetic energy of a body in motion: `(mv2)/2`
• The area of a circle: `(Tr2)/2`

In fact, those formulas can all be derived in exactly the same way, using Integral Calculus. So if someone wants to argue that π is the correct constant to use (τ/2), then they should also be arguing that the correct gravitational constant is `g/2`, rather than `g`!

That is a ridiculous argument of course. But so is the notion that π is the “superior” constant, when it is clearly an out of step with similar mathematical constructs, in addition to being an impediment to understanding.

## References

1. January 12, 2021 (7:28 am)

Try or imagine this: cut a piece of string to the length of the diameter of a circle. Then see how many times that string can wrap around the circumference. It will be 3 times plus a little extra. That little extra is the .14159… of pi. This worked really well when I taught 5th grade and equally well when I taught adult GED. This becomes their anchor: there are little more than 3 diameters out on the circumference. They never forget this if they’ve actually done the measuring, cutting, laying of string, and marking that string as they go. They also hang on to the .14 of pi very well. They don’t just think, 3 point something… because we would actually know decimals well at that point and we’d all estimate what fraction (in decimal form) of string it took for that little bit left.
So then we start with circumference as diameter x pi and get really good at that before bringing in the easy idea that there are two r’s in the diameter, to be flexible when you see it written as 2r•pi or 2pi•r.
Since 2•pi=tau, I don’t think it would be all that helpful in terms of circumference since we’re really dealing with the r and the d and their relationship: instead of pi•2, we are using r•2 which is d. And we started with d with the string thing, so that’s our anchor.

• January 12, 2021 (8:14 am)

Really great teaching device, Paul. Thanks for sharing it.

I love that you put 2 r’s back to back to identify one d.
That’s nice. Then you focus on 2r = d, rather than thinking
about 2 * pi. That’s nice, too.

I also agree that it’s harder to go the other way, so your
teaching sequence makes a lot of sense!

At that point, though, I’d love to find a way to transfer the
focus to Tau. It’s just a whole lot more useful in real life
to think about Tau as a “turn”!

If I have a pipe with a given radius, how long is the tape that
will go around it? Tau gives me the answer in a trice, because
I know it is a “turn”. Do I remember that for 2*pi? Never did!
:_)

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