Radians: the natural way of measuring angles
Another one for Wikipedia. Tumblr forced me to cut the amount of frames in half. Here it is in its full, smooth glory.
However, only one of these angle units earns a special place in mathematics: the radian.
This animation illustrates what the radian is: it’s the angle associated with a section of a circle that has the same length as the circle’s own radius.
For a unit circle, with radius 1, the radian angle is the same value as the length of the arc around the circle that is associated with the angle.
In the animation, the radius line segment r (in red) is used to generate a circle. The same radius is then “bent”—without changing its length—around the circle it just generated. The angle (in yellow) that’s associated with this bent arc of length r is exactly 1 radian.
Making 3 copies of this arc gets you 3 radians, just a bit under half of a circle. This is because half of a circle is π radians. So that missing piece accounts for π - 3 ≈ 0.14159265… radians.
Our π radians arc is then copied once again, revealing the full circle, with 2π radians all around.
There are several great reasons to use radians instead of degrees in mathematics and physics. Everything seems to suggest this is the most natural system of measuring angles.
Radians look complicated to most people due to their reliance on the irrational number π to express relations to circle, and the fact the full circle contains 2π radians, which may seem arbitrary.
In order to simplify things, some people have been proposing a new constant τ (tau), with τ = 2π. When using τ with radians, fractions of τ correspond to the same fractions of a circle: a fourth of a tau is a fourth of a circle, and so on.
Tau does seem to make more sense than pi when dealing with radians, but pi shows up elsewhere too, with plenty of merits of its own.
I, for one, do enjoy the idea of tau being used, exclusively, as an angle constant, so that it immediately implies the use of radians. If such were the case, a student seeing Euler’s identity for the first time, but in terms of tau, would be immediately compelled to think in terms of rotations: eτi = 1 would instantly convey the idea of a full rotation, bringing you back where you started. That seems like a good thing.
So happy Pi day!
(or half-tau day, if you prefer!)