Definition:Angular Measure/Degree

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The degree (of arc) is a measurement of plane angles, symbolized by $\degrees$.

\(\ds \) \(\) \(\ds 1\) degree
\(\ds \) \(=\) \(\ds 60\) minutes
\(\ds \) \(=\) \(\ds 60 \times 60 = 3600\) seconds
\(\ds \) \(=\) \(\ds \dfrac 1 {360}\) full angle (by definition)

Value of Degree in Radians

The value of a degree in radians is given by:

$1 \degrees = \dfrac {\pi} {180} \radians \approx 0 \cdotp 01745 \, 32925 \, 19943 \, 29576 \, 9236 \ldots \radians$

This sequence is A019685 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Also known as

Degrees of arc are also referred to as sexagesimal measure.

See sexagesimal notation for further explanation.

Also see

  • Results about degrees of arc can be found here.

Historical Note

The division of the circle into $360$ degrees originates from the Babylonians, who used a sexagesimal (base $60$) number system for the purposes of mathematics and astronomy.

Degrees are usually the first way of measuring angles taught to mathematics students, usually at grade school.

Conveniently, the most commonly used angles in geometry (for example $30 \degrees$, $45 \degrees$, $60 \degrees$) are all whole numbers when measured in degrees.

Technical Note

The $\LaTeX$ code for \(\degrees\) is \degrees .