# Radian

A radian (sometimes indicated as "rad") is a unit of measurement for angles. It is used in many areas of mathematics, such as trigonometry, calculus and, more.

## Radian definition

A radian is a measurement of angle based on the radius of a circle. 1 radian is the angle that is subtended by an arc that has a length equal to the radius of the circle. In the figure below, θ=1 radian.

The radian measure of any central angle (angle whose vertex is at the center of a circle) is equal to the length of the intercepted arc divided by the radius, or radians, where θ is the angle in radians, s is the arc length, and r is the radius.

As a ratio of two lengths, radians are commonly considered a "pure number," meaning that it is dimensionless. Thus, "rad" is very rarely used to indicate that an angle measure is in radians. Rather than writing "1 rad," a radian is most typically written as simply "1" and an angle measure that doesn't specify some other unit is assumed to be in radians.

## Radian v.s degree

Degree is another unit of measure for angles, denoted by the symbol °. An angle measuring 1° is equal to of one complete revolution of the angle about its vertex.

To convert from radians to degrees, or degrees to radians, use the following conversions:

Thus, to convert a measure that is in degrees to radians, multiply by . To convert an angle in radians to degrees, multiply by .

When using degrees, a degree (°) symbol is used to indicate that the angle is not in radians.

1 radian ≈ 57.296°

## Radians in a full circle

The circumference of a circle is 2πr where r is the radius of the circle. Since the circumference of a circle encompasses one complete revolution of the circle, its arc length is s = 2πr. Plugging this into the formula for radian measure,

and 2π ≈ 6.28, so there are approximately 6.28 radians in a circle:

## Commonly used angles in radians

We can use the fact that 1 complete revolution equals 2πr to find angle measures in radians and degrees that are commonly used in the study of trigonometry.

1 revolution = 2π rad = 360° |

of a revolution = π rad = 180° |

of a revolution = π rad = 120° |

of a revolution = π rad = 90° |

of a revolution = π rad = 60° |

of a revolution = π rad = 45° |

of a revolution = π rad = 30° |

Below is a list of angles on a circle measured in degrees and radians that are commonly used in trigonometry.