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 $\degrees$.

\(\ds \)

\(\)

\(\ds 1\)

degree

\(\ds \)

\(=\)

\(\ds 60\)

minutes

\(\ds \)

\(=\)

\(\ds 60 \times 60 = 3600\)

seconds

\(\ds \)

\(=\)

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

full angle (by definition)

Minute

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

\(\ds \)

\(\)

\(\ds 1\)

minute

\(\ds \)

\(=\)

\(\ds 60\)

seconds

\(\ds \)

\(=\)

\(\ds \dfrac 1 {60}\)

degree of arc (by definition)

Second

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

\(\ds \)

\(\)

\(\ds 1\)

second

\(\ds \)

\(=\)

\(\ds \dfrac 1 {60}\)

minute of arc (by definition)

\(\ds \)

\(=\)

\(\ds \dfrac 1 {60 \times 60} = \dfrac 1 {3600}\)

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.