Radian Measure: Definition & Formula

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  • 0:02 Definition of Radian
  • 0:58 Convert Degrees to Radians
  • 1:39 Example Problems
  • 2:44 Advantages of Using Radians
  • 3:25 Lesson Summary
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Lesson Transcript
Instructor: Jennifer Beddoe

Jennifer has an MS in Chemistry and a BS in Biological Sciences.

In mathematics, the radian is the standard unit of angular measure. This lesson will define radian and work through some problems involving radians. The lesson will end with a quiz.

Definition of Radian

Radians are the standard mathematical way to measure angles. One radian is equal to the angle created by taking the radius of a circle and stretching it along the edge of the circle.


The radian is a pure mathematical measurement and, therefore, is preferred by mathematicians over degree measures. For use in everyday work, the degree is easier to work with, but for purely mathematical pursuits, the radian gives better results. You probably will never see radian measures used in construction or surveying, but it is a common unit in mathematics and physics.

The concept of measuring an angle by the length of the arc was first made popular in the early 1700s by Roger Cotes, an English mathematician who worked closely with Isaac Newton, but the term radian was not used to describe this concept until the late 1800s by James Thomson at Queen's College, Belfast, Ireland.

Convert Degrees to Radians

The unit used to describe the measurement of an angle that is most familiar is the degree. To convert radians to degrees or degrees to radians, the following relationship can be used:

angle in degrees = angle in radians * (180 / pi)

This equation works because a half circle is 180 degrees and is also equal to pi radians.

180 degrees = pi radians

1 radian = 180 / pi

This chart also is helpful in converting degrees to radians or radians to degrees. It also gives a good overview of the radian and how it is used to measure angles.


Example Problems

Let's look at some examples to help you practice converting degrees and radians.

Convert 45 degrees to radians

  • 45 = radians * (180 / pi)
  • 45 = 57.32 * radians
  • radians = 45 / 57.32
  • radians = 0.785

Most often, when writing degree measure in radians, pi is not calculated in, so for this problem, the more accurate answer would be: radians = 45 pi / 180 = pi / 4.

Convert pi/3 radians to degrees

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