I see. That almost makes sense, but pi radians = 180°

Also, the value of one internal angle of a regular polygon is (n-2)×(π÷n), in which case π÷n is infinitesimally small. In other words, substituting infinity for n would be incalculable, and even if it were, adding them together wouldn't equal infinity because the larger n is, the smaller each individual internal angle.

It's not about colloquialism or language, there are immutable principles of geometry, and adding the internal angles of a triangle gives you 180°, whether you express it as such or as π radians or 3200 mils or something completely different doesn't matter. That's just changing the unit of measurement but the underlying principle is the same.

Circles can be confusing and counterintuitive, but that's why they need an irrational number in order to be expressed. If you're measuring the internal angle you'll probably express it as an arc, because infinite and infinitesimal numbers are impossible to express rationally.

Take for instance, calculating angular momentum with a circle. You have to calculate it based on the tangent because the circle itself doesn't give you any constancy otherwise.