Radian - Math Word Definition - Math Open Reference

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Math Open Reference Home Contact About Subject Index Radian A unit of measure for angles. One radian is the angle made at the center of a circle by an arc whose length is equal to the radius of the circle. Try this Drag the orange dot. Note that the radian is a fixed angle no matter what the size the circle.

The radian is a unit of measure for angles used mainly in trigonometry. It is used instead of degrees. Whereas a full circle is 360 degrees, a full circle is just over 6 radians.

A full circle has 2π radians (Roughly 6.28) As seen in the figure above, a radian is defined by an arc of a circle. The length of the arc is equal to the radius of the circle. Because of this the radian is a fixed size no matter what the size of the circle is. Drag the orange point and convince yourself this is so.

Recall that the circumference of a circle is 2πR, so that means there are 2π, or roughly 6.28 radians in a full circle. Because a full circle is also exactly 360°, each radian comes out to approximately 57.296°.

To convert between degrees and radians

Given an angle whose measure is D degrees or R radians:
Exact Approximate
Given degrees (D) 0.017D radians
Given radians (R) 57.3R degrees

Degrees / radians converter

In the boxes below, enter either the number of degrees, or the number of radians (not both) and click on 'convert'.
Degrees:
Radians:
Convert

Common angles in degrees and radians

One of the reasons for using radians is that common angles come out to be convenient fractions of pi, as you can see:
Exact radians Approximate radians
30° 0.52
45° 0.785
60° 1.048
90° 1.57
180° 3.142
270° 4.71
360° 6.28

Other angle topics

General

  • Angle definition
  • Degrees
  • Radians
  • Angle bisector
  • Subtended angle
  • Interior of an angle
  • Included angle

Angle Types

  • Acute angle
  • Right angle
  • Obtuse angles
  • Straight angle
  • Reflex angle
  • Full angle

Angle relationships

  • Vertical angles
  • Complementary angles
  • Supplementary angles
  • Linear pair
  • Adjacent angles
  • Corresponding angles
  • Alternate interior angles
  • Alternate exterior angles
  • Interior angles of a transversal
  • Exterior angles of a transversal
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