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