Definition:Angle/Unit

Definition

The usual units of measurement for angle are as follows:

Degree

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

\(\displaystyle \)

\(\)

\(\displaystyle 1\)

$\quad$ degree

$\quad$

\(\displaystyle \)

\(=\)

\(\displaystyle 60\)

$\quad$ minutes

$\quad$

\(\displaystyle \)

\(=\)

\(\displaystyle 60 \times 60 = 3600\)

$\quad$ seconds

$\quad$

\(\displaystyle \)

\(=\)

\(\displaystyle \dfrac 1 {360}\)

$\quad$ full angle (by definition)

$\quad$

Minute

The minute (of arc) is a measurement of plane angles, symbolized by $'$.

$1$ minute $= 60$ seconds (of arc) $= \dfrac 1 {60}$ of a degree of arc.

Second

The second (of arc) is a measurement of plane angles, symbolized by $''$.

$1''$ is defined as $\dfrac 1 {60}$ of a minute of arc, or $\dfrac 1 {3600}$ of a degree of arc.

Radian

The radian is a measure of plane angles symbolized either by the word $\radians$ or without any unit.

Radians are pure numbers, as they are ratios of lengths. The addition of $\radians$ is merely for clarification.

$1 \radians$ is the angle subtended at the center of a circle by an arc whose length is equal to the radius:

Also see

• Pi ($\pi$)

• Full Angle measures $2 \pi$ Radians

In the context of calculus in particular, it is usual to use radians.

This is because then the derivatives of trigonometric functions, such as, for example:

$\dfrac {\mathrm d} {\mathrm d x} \sin x = \cos x$

work out without the necessity of multiplicative constants.