Full Angle measures 2 Pi Radians

Theorem

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

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

This sequence is A019692 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

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.

$\blacksquare$

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $6 \cdotp 283 \, 185 \ldots$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $6 \cdotp 28318 \, 5 \ldots$