Value of Radian in Degrees

Theorem

The value of a radian in degrees is given by:

$1 \radians = \dfrac {180 \degrees} {\pi} \approx 57 \cdotp 29577 \, 95130 \ 82320 \, 87679 \, 8154 \ldots \degrees$

This sequence is A072097 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Proof

By Full Angle measures 2 Pi Radians, a full angle measures $2 \pi$ radians.

By definition of degree of arc, a full angle measures $360$ degrees.

Thus $1$ radian is given by:

$1 \radians = \dfrac {360 \degrees} {2 \pi} = \dfrac {180 \degrees} {\pi}$

$\blacksquare$

Also see

• Value of Degree in Radians

Sources

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): Table $1.1$. Mathematical Constants
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 1$: Special Constants: $1.26$
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.13$
• 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): $57 \cdotp 296 \ldots$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $57 \cdotp 296 \ldots$