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 &amp;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-conformist2, this is actually part of a world-wide movement to correct the circle constant to what it should have been.

1. In fact, what is the diameter but $$2r$$ anyway? ↩︎