Pi and Tau

Many people believe π is wrong, and here's why. Tomorrow is Pi Day, so in honor of that, I will slander π! Let's begin with a bit of background knowledge. π is equal to the circumference of a circle divided by its diameter. It's equal to about 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998.* But there is a growing movement to use a number called tau instead, denoted by the Greek letter τ. τ is equal to the circumference of a circle divided by its radius. τ is equal to 2π, or about 6.2831.** τ can be a more logical choice than π for the circle constant. The primary argument for τ has to do with radians. The system of measuring angles with degrees is rather arbitrary. Why should a circle be 360 degrees? So, in higher math, radians are used instead. Here's how radians are defined: in a circle with radius 1, the measure of an angle is equal to the length of the arc across from it. Let's look at some sample angles. (From the Tau Manifesto.) A 360° angle spans the entire circle. To calculate its circumference, we take π times the diameter, which is 2, since the radius is 1. This gives us 2π. For 180°, or 1/2 of a circle, we get π.  For 120°, or 1/3 of a circle, we get 2π/3.  For 90°, or 1/4 of a circle, we get π/2.  For 60°, or 1/6 of a circle, we get π/3. For 45°, or 1/8 of a circle, we get π/4. For 30°, or 1/12 of a circle, we get π/6. Do you see the problem? There is an errant factor of 1/2 floating around here that causes all of the radian measures to fall askew. Where did this come from? Well, π is defined based on the diameter, and our circle has a radius of 1 … and a diameter of 2. How do the tauists combat this problem? Simple. Use τ. After all, τ is defined based on the radius, which is more widespread than the diameter anyhow.

(Still from the Tau Manifesto.) A 360° angle spans the entire circle. To calculate its circumference, we take τ times the radius, which is 1. This gives us τ. For 180°, or 1/2 of a circle, we get τ/2. For 120°, or 1/3 of a circle, we get τ/3. For 90°, or 1/4 of a circle, we get τ/4. For 60°, or 1/6 of a circle, we get τ/6. For 45°, or 1/8 of a circle, we get τ/8. For 30°, or 1/12 of a circle, we get τ/12. It all makes sense now.*** Now if you're a tauist****, you might stop here. Maybe throw in a snappy bit about the Euler Formula. But π is not without its defenders. Here's their convincing graphic:

(From the Pi Manifesto.) π makes the area of a circle much simpler to calculate. Now, if you're a devout tauist, you might cook up some argument about how the 1/2 in that area is meant to be there, because of triangles or something. But let's be honest: π wins this one. It is simply more intuitive to calculate the area of a circle using π.  Another argument on the side of π is that it is easier to measure the diameter of a circle, and looks more visually appealing. (See this video.) You might be wondering, though, why does this actually matter? There might be a few extra 2s in some formulas. Mathematicians can deal. But this is not about mathematicians. This is about people who are just learning about radians, people who are confused by that factor of 2, and don't understand. Elegance is important in math.  I'm just saying, whether you support pi, tau, or pau, you should know the facts.***** Thanks for reading what I wrote. Remember that τ will always be greater than π. –beautifulthorns *Sorry, I memorized a lot when I was younger. **I know, it's quite a disappointment.  ***To be clear, changing π to τ is not the only way to solve this problem. We could also use the circle with diameter 1. This new unit is called a darian. If you're a π person, this is one way to rectify the radians issue.  ****I couldn't find a good place to say this, so I'm putting it in a random footnote. Check out Vi Hart's videos on this topic. *****I am myself a tauist, so I'm sure the post is slanted that way. But I tried.