[rvbissell]

Actually, no: that Tau-centric area formula you gave derives naturally from taking the integral. Your example actually fits the expectation you have from what you learned in Calculus I. You should _expect_ that 1/2 scaling to be there.

If it seems awkward to you, it's only because of a lifetime of seeing it done in terms of pi.

[mistercow]

You can argue that having an extra number to juggle around is somehow less awkward because, under specific and subjective criteria, it’s “expected”, but given that the whole hook for tau is “we keep having to put a multiplier of two everywhere”, I don’t find it very compelling.

I can also make arguments that pi/2 would have been a better constant from a teaching perspective. The pi/2 version of the Euler identity, for example, would give you all the tools you need to link complex multiplication to rotation.

But at the end of the day, the choice of multiple used for the constant is a convention. None is going to be ideal in every case, and no math fundamentally changes because of a particular choice. Trying to argue for a change in convention at this point is just silly.