If your natural unit is radians, then you're arguing for tau rather than against it.

How many radians in the circle? Tau

How many in half a circle? Tau / 2

How many in a quarter turn? Tau / 4

Much better than mentally switching from a quarter circle to half a pi.

And if you're using it in calculus, it's more natural to integrate from 0 to Tau than from 0 to 2 Pi.

Read the Tau Manifesto (linked int he article). He covers that and shows why Tau is easier to use there, too. Dude was a physics prof at Caltech; I think calculus is sort of important to him, too.

Wait..

1. Calculus was invented to make the geometry easier -- basically opposing geometry and calculus is a broken idea.

2. Geometers use radians too.

3. All the stuff other repliers said about 1 tau radians...

Then why the full angle is 2 dammed Pi radians? I prefer the simpler Tau radians instead.