In the lead-up to Tau day, I’ve been thinking on how our existing formulas for circles and spheres are poorly served. They are inconsistent and confusing, which is something that is easily fixed when we use Tau instead of Pi.

Here’s how Tau helps us to better understand the formulas for circles and spheres.

First up, a comparison of some standard formulas using Pi, 2 Pi, and Tau.

\pmb{\pi} \pmb{2\pi} \pmb{\tau}

Circles

Circumference C \pi d 2\pi r \tau r
Area A \pi r^2 \frac{1}{2}2\pi r^2 \frac{1}{2}\tau r^2

Spheres

Area A 4 \pi r^2 2(2\pi) r^2 2 \tau r^2
Volume V \frac{4}{3} \pi r^3 \frac {2}{3} 2\pi r^3 \frac{2}{3} \tau r^3

Overall, we can see that just using 2 \pi doesn’t provide enough of an improvement. To gain the real benefits you need to go all the way to using Tau (\tau=2\pi).

Circumference

\pi d 2\pi r \tau r

The traditional formula with \pi d is inconsistent in using diameter, and only serves to disguise that there’s a factor of two in there. With \tau r it becomes clear that Tau is the circle constant.

This becomes even more apparent when measuring the circumference of a unit circle, which is used all the time with sine and cosines and are measured in radians.

Pi angles of radians

Pi angles of radians

With the Pi angles of radians, it becomes clear when comparing it with the below image that Pi is only half of the story.

Tau angles in radians

Tau angles in radians

The Tau angles make so much more sense. A quarter of the circle is \frac{1}{4}\tau, half the circle is \frac{1}{2}\tau, and so on. It doesn’t get much easier to understand than that.

Circle Area

\pi r^2 \frac{1}{2}2\pi r^2 \frac{1}{2}\tau r^2

If Tau is the circle constant and we’re complaining about doubling Pi, then why here are we halving Tau? Let’s unwrap the circle and find out why.

TriangleFromCircle

The half comes from the area of the triangle, which is a half base times height. With a circle, the half just-so-happens to cancel with the 2 from 2 Pi.

There are all sorts of other formulas that we use too, where slices are added up to find the whole.

Distance fallen y=\frac{1}{2}gt^2
Spring energy U=\frac{1}{2}kx^2
Kinetic energy K=\frac{1}{2}mv^2

And there are many others too, such as angular displacement, rotational energy, potential energy, electric field energy density, energy in a capacitor, and the list goes on.

All of these are where we’re finding out how something changes (such as gt) when we progressively change something (such as t). The same thing happens with the circumference of a circle in relation to its radius too. Wikipedia has a lot more detail at the triangle method for the area of a disk too.

When the circle constant Tau is kept consistent throughout, the formula for the area then becomes consistent with many other related ones too.

Sphere Area

4 \pi r^2 2(2\pi) r^2 2 \tau r^2

What does the 4 relate to and how does it help us to understand the sphere area?

With Tau, we can see that the surface area of the top half of a sphere is \tau r^2, which along with the bottom half of the sphere, add up to two physical halves of the sphere. That being 2\tau r^2.

Sphere Volume

\frac{4}{3} \pi r^3 \frac {2}{3} 2\pi r^3 \frac{2}{3} \tau r^3

The 4 in \frac{4}{3} \pi r^3 doesn’t seem to tell us anything useful about the sphere.

When using Tau, we can easily relate this to Archimedes who famously found that when we have a cylinder around a sphere, the sphere is two thirds the volume of that cylinder. More details on this can be found at places such as spheres in cylinders.

With the two-thirds part of a sphere, I find that this helps to form a much stronger bond with Archimedes and the roots of how such things were discovered.

Sphere cylinder, from Tomb of Archimedes

Sphere cylinder, from Tomb of Archimedes

 

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