Measurements of Common Angles/Full Angle

Theorem

A full angle is equal to $360 \degrees$ or $2 \pi$ radians.

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.

By definition of degree of angle, a full rotation is $360 \degrees$.

Therefore, a full angle is $360 \degrees$ or $2 \pi$.

$\blacksquare$

Sources

• 1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $1$. Functions: $1.5$ Trigonometric or Circular Functions: $1.5.1$ Unit Circle
• 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): $6 \cdotp 28318 \, 5 \ldots$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): angle
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): angular measure
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): angle
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): angular measure
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): angle (angular)