Every time I saw a Pi (\( \pi \)) symbol in mathematics that represented the value 3.141592653…, there was usually a “2” in front of it. This is no coincidence: a circle is the set of points that are the same distance \( r \) from a center point, where \( r \) is the radius, so the *radius* is what defines a circle—not the *diameter*. ^{1}

Now, the value of \( \pi \) represents **the ratio of a circle’s circumference to its diameter**… wat? No wonder there’s always a persistent factor of 2 in almost every equation involving \( \pi \): The proper circle constant should be the ratio of a circle’s circumference to its

*radius*, or \( 2\pi \approx 6.283185307\ldots = \tau \) (see the figure below). This has a profound connection with radians and the trigonometric functions sine, cosine, tangent, etc. Even the Euler Formula looks better: \( \mathrm{e}^{i , \tau} = 1 \).

```
<img src="http://upload.wikimedia.org/wikipedia/commons/2/28/Circle_radians_tau.gif" class="figure-img img-fluid rounded" alt="There are &tau; radians in a circle">
<figcaption class="figure-caption">
<cite><a href="http://commons.wikimedia.org/wiki/File%3ACircle_radians_tau.gif" target="_blank">Image from Wikimedia Commons</a></cite>
</figcaption>
```

This is not just me being non-conformist^{2}, this is actually part of a world-wide movement to correct the circle constant to what it should have been.

Read more about why pi is wrong at tauday.com