Pi is Stupid
I teach Brazilian Jiu-Jitsu to people of all ages, from preschoolers to middle-aged parents. While BJJ, in itself, is not necessarily the most academic of pursuits, I also happen to be a huge nerd. So while teaching some of my 8 to 13 year-old students, it sometimes happens that I overhear them talking about their math classes, often to complain about ideas that they’re struggling to grasp. Being a huge nerd, and also a delighted teacher, I do my best to help them through these issues. If I can teach a kid how to find the length of the hypotenuse of a right triangle at the same time as teaching her how to finish a Triangle Choke, I become pretty much the proudest martial arts instructor you could hope to meet.
One of the things that my kids often use in their math classes, but almost never really understand, is the constant π (pi). They are taught π in class to help learn things like how to calculate the area of a circle, but they usually don’t really know what π actually is. They just think of it as some number that they have to memorize, never thinking about where the number comes from, or why it is what it is. Sometimes, I’ll tell the kids that they can earn their way out of doing push-ups if anyone can tell me what π is. Most often– after the jokes about desserts are made– I’ll hear someone say, “Coach, π is three-point-one-four!” Every now and again, one of the kids is clever enough to say, “Coach, π is three-point-one-four-on-into-infinity!” They get confused when I tell them that’s the value of π, but that is not what π actually is. It’s not their fault that they get confused by this; they were usually taught about π all wrong. I don’t even blame their math teachers, because most of the time, those math teachers were also taught about π in the wrong way. For a very long time, math classes have been teaching that π is a number, instead of teaching that π is the relationship between a circle’s circumference and its diameter. There is a reason it has been taught this way.
Ladies and gentlemen, π is just plain stupid.
Let me explain, for a moment. I don’t mean that the value of π is incorrect, and I don’t mean that π is not actually the ratio of a circle’s circumference to its diameter. What I mean is that thinking about circles in terms of π is counterintuitive. Even once they learn geometry, most people still don’t understand why we use π in our calculations; they simply know that we use it. The reason for this strange disconnect, in my opinion, is that mathematics describes circles by their radius in almost every single instance; except for the definition of π, where we instead look at the circle’s diameter. This is sort of like always measuring a racetrack in kilometers, except when it is a motorcycle track, in which case you measure in miles. Sure, you can convert miles to kilometers easily enough, but does it really make sense to do things this way?
The best way to explain what I mean will probably be to illustrate the issue. After all, circles are very visual things, and it can be difficult to really understand what I’m trying to say unless you can see it in front of you. So, let’s start by defining exactly what a circle actually is. A circle is the set of all points in a plane which are equidistant from a single origin. Look at the diagram, below. The curved, green line is our circle. At the center of the circle is point O, the origin point. Points A and B lie on the circle a distance r from the origin, just like every other point on the circle. We refer to this distance r as the radius.
Now, let’s pretend that we have a farmer who owns a small chunk of land. According to his deed, the farmer owns all the land within one mile of his farmhouse. As we can see by our definition, this means that the boundary of the farmer’s land is a circle with a 1 mile radius which has an origin point at the farmhouse. Now, the farmer needs to plan how much seed he will plant, and how to irrigate his new land; and in order to do that, the farmer needs to know exactly how much land he will actually be working. He needs to know the area of the circle.
If I presented this problem to my kids, they would joyfully exclaim, “Simple, coach! The area of a circle is !” Unfortunately, being the meanie that I am, I would then ask the kids, “But why is the area of a circle ?” Let’s pretend, for a moment, that our farmer has never seen the formula for the area of a circle, before. He can do some simple arithmetic, but he’s no mathematician. So, he decides to take this problem over to his friend, Archie, who happens to be a really well-renowned math scholar. He then asks Archie how to find out how much land his farm is going to cover.
Archie thinks about the problem for quite a while. It’s easy to find areas for shapes which are bounded by straight lines, like triangles and rectangles, but circles are very difficult. So Archie decides he wants to compare the area of the circle to one of these easier areas. He draws the following picture, and asks a simple question: does the right triangle have a smaller area than the circle, a bigger area, or the same area?
Let’s call the area of the circle and the area of the triangle . Now, Archie still doesn’t know how to find , but he knows that the area of a triangle is where is the length of the triangle’s base and is its height– because a triangle is half of a rectangle, and a rectangle’s area is its length times its height. So that means our triangle, in Figure 2, has an area of . Now, here’s where he gets really clever. Archie draws a triangle inside the circle, with one vertex at the origin and two on the circle itself. If he copies this triangle, rotating the copies inside the circle, he can make a new shape called a polygon. And since he can calculate the area of each of those little triangles, he can calculate the area of that whole polygon, , and compare it to . If the area of the right triangle, , is less than the area of the circle, , then we should be able to make a polygon inside the circle which has an area equal to .
The area of this polygon is given by the formula where is the number of triangles. However, multiplying the base length of each triangle, , by the number of triangles, , simply gives us the perimeter of the polygon, . This means that our formula becomes . We can see, in Figure 3, that no matter how many triangles we make inside the circle, the perimeter of our polygon will be smaller than the circumference of the circle () and the height of the triangles will be less than the radius of the circle (), so we can conclude that . No matter how many sides the polygon has, its area will still be smaller than . It is impossible to create a polygon inside the circle which has an area equal to our right triangle. This means that the right triangle from Figure 2 cannot possibly have a smaller area than the circle.
Is it possible, then, that is greater than ? We can perform another experiment with triangles in order to find out. Let’s say we make an isosceles triangle with height which has one vertex at the origin of the circle and two vertices which lie outside the circle, such that the base of the triangle is tangent to the circle at its midpoint. We can similarly rotate this triangle about the origin to form another polygon. If the area of the right triangle is bigger than the area of the circle, then we should be able to create a polygon surrounding the circle which has an area equal to . Again, we know that ; but in this case, we know that the perimeter will always be larger than the circumference () and that the height is equal to the radius (). Therefore, we know that , no matter how many triangles we use to construct the polygon. It is impossible to create a polygon surrounding the circle with an area equal to our right triangle. This means that the right triangle from Figure 2 cannot possibly have a bigger area than the circle.
So, now we have determined that cannot be smaller than the area of the circle, nor can be larger than . Therefore, we must conclude that ; that is to say, the area of a circle with radius and circumference is equal to the area of a right triangle with legs of length and . So, that means that . Of course, we still have a problem: we don’t know the value of , nor a method for calculating it.
Now, you’re probably saying to yourself, “That’s great, but what the heck does any of this have to do with π?!” Well, our Archie was actually Archimedes of Syracuse, an extremely smart mathematician. He recognized this problem with calculating the circumference, as well, and he actually discovered a pretty brilliant method to help solve the problem, too, using inscribed and circumscribed polygons, like the ones from Figures 3 & 4. Unfortunately, when he formulated his conclusion, Archie stated, “The ratio of the circumference of any circle to its diameter is less than but greater than .” This was the first rigorous attempt to calculate π in mathematics history, but did you notice the switch? Throughout this entire discussion, we defined and described the circle entirely by its radius. However, when we finally got to the conclusion, Archimedes decided to put it in terms of the diameter– which is 2 times the radius! We can speculate any number of reasons why Archie might have done this, but the unfortunate fact of the matter is that mathematicians spent the next 2000 years following his lead. Circle functions were always described by the ratio of the circle’s circumference to its diameter, even though the radius is a much more natural value for comparison. Every time you use π in an equation, that equation has to account for the conversion between radius and diameter. This is a terrible and fundamental mistake which has been making mathematics more difficult for an extremely long time.
But don’t worry, friends. There is a better way!
What if we decided to define our circle constant by the radius instead of the diameter? Let’s call the ratio of a circle’s circumference to its radius τ (tau), since it represents one full turn (Greek, τορνος) of a circle. Way back in the beginning of this article, we asked why the area of a circle is . Even after all the work we’ve done, it still might not be readily apparent. So, let’s switch tack and look at τ, instead. What if I told you that the formula for the area of a circle is ? Suddenly, it’s much easier to see why this is true. We already proved that . We have defined our new circle constant as , but if we solve this equation for we find that . Plugging this back into our area formula, we get ! It’s really easy to see why the area formula is what it is, using τ; as opposed to using π, where it gets totally obfuscated.
So, let’s look at a few areas where τ is superior to π…
Circumference of a Circle
When we use τ in the formula for the circumference of a circle, we show a simple, direct relationship between the circumference of the circle and its primary property, the radius. On the other hand, when using π, we either have to define in terms of diameter, which is a property of circles that is almost never utilized in mathematical formulae, or we have to add a second corrective constant to our equation, the number 2, in order to convert π into a useful value. Why not just employ a useful value, to start with, instead of a stupid one?
Area of a Circle
At first glance, some people might think π is the better choice, here, since the final equation is simpler. However, as we showed in the main article above, this actually makes it more difficult for people to understand why the area formula is as it is. It becomes a matter of memorization, rather than one of actual comprehension. This would not be so bad if this was the only equation where this was the case, but unfortunately the area formula sets a precedent of memorization-over-understanding which is then inherited by almost every other formula involving circles– and there are a lot of other formulae which involve circles. The use of τ resolves this problem by reminding the student that a circle has the same area as a right triangle with legs of length and . Less memorization makes for smarter students.
Do you remember how confusing it was to have to memorize all those angles in radians? When you couldn’t remember what angle was supposed to be representing, you’d convert it back to degrees, first, then you knew that 120° is a bit more obtuse than a right angle. So weird! But, if you use, τ you see that the angle is , which is just one third of a circle.
Look at the image, above. Using τ makes angular measurement so simple! You don’t have to worry about dividing the circle up into 360 degrees– an archaic legacy of ancient Babylonian mathematics which still persists, today, for some stupid reason– nor do you have to memorize the strange formulations with π that constantly confuse trigonometry students. Want to know the angular measurement of an eighth of a circle? That’s just . Looking at a right angle? Well, a right angle is just a quarter of a circle– . Trying to figure out what angle radians represents? Well, that’s just of a circle! It is immediately obvious that τ is superior to both π and degrees of angle, when one looks at Figure 5.
Look at how much more naturally τ measures the Sine wave than does π! Just like we saw in the angular measurement, above, it is much easier to discuss the period of trigonometric waves in terms of τ, which makes these things far easier for teachers to teach and for students to understand. The sine wave measures properties of a unit circle. The maximum value of is the radius , and the period of the wave is the circumference . This makes far more sense when we are using a circle constant which is directly in terms of and , like τ, than one which is in terms of and , like π.
As we can very easily see, , therefore τ is greater than π. ‘Nuff said!
More than two thousand years ago, a brilliant mathematician made a dumb mistake when he was trying to figure out some of the crazy aspects of circles. Unfortunately, almost every mathematician that followed him just figured they’d follow his lead, without thinking that there might be a better way to talk about circles. It wasn’t until relatively recently that a small handful of people began to realize just how stupid π actually is. Back in 2001, mathematician Bob Palais published an article in The Mathematical Intelligencer called Pi is Wrong! in an attempt to bring this issue to light. Some time later, physicist Michael Hartl published his Tau Manifesto, which became an internet sensation and really began the movement which eventually showed me the error of Archie’s ways (and which was the source for the images in Figures 5 & 6). If you want to see an in-depth look at even more ways in which τ is superior to π, I definitely recommend Hartl’s page. Since the publication of the Tau Manifesto, the case for τ has been picked up by educators, students, and general nerds all over the world, and while our effort is still slow going, we have definitely made some progress in alerting people to the fact that π is stupid.