# Full Angle measures 2 Pi Radians

## Theorem

One full angle is equal to $2 \pi$ radians.

$2 \pi \approx 6 \cdotp 28318 \, 53071 \, 79586 \, 4769 \ldots$

## Proof

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

From Perimeter of Circle, the length of the circumference of a circle of radius $r$ is equal to $2 \pi r$.

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

It follows that $2 \pi$ radians sweeps out the entire circle, or one full angle.

$\blacksquare$