Full Rotation is 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 $\dfrac 1 {2 \pi}$th of a circle.

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

$\blacksquare$