Full Angle measures 2 Pi Radians

Theorem
One full rotation is equal to $$2\pi$$ radians.

Proof
From the definition of radians, 1 radian is the angle which sweeps out an arc on a circle whose length is the radius $$r$$ of the circle.

From the definition of pi, the circumference $$C$$ of a circle is equal to $$2\pi r$$.

Therefore, 1 radian sweeps out $$\frac{1}{2\pi}$$th of a circle.

It follows that $$2\pi$$ radians would sweep out the entire circle, or one full rotation.