Conversion from Degrees to Radians Help

By — McGraw-Hill Professional
Updated on Oct 4, 2011

Introduction to Conversion from Degrees to Radians

Trigonometry has been used for over two thousand years to solve many real-world problems, among them surveying, navigating, and problems in engineering. Another important use is analytic—the trigonometric functions and their graphs are important in several mathematics courses. The unit circle is the basis of analytic trigonometry. The unit circle is the circle centered at the origin that has radius 1. See Figure 13.1.

Angles have two sides, the initial side and the terminal side. On the unit circle, the initial side is the positive part of the x-axis. The terminal side is the side that rotates. See Figure 13.2.


Fig. 13.1


Fig. 13.2

A positive angle rotates counterclockwise, Trigonometry. A negative angle rotates clockwise, Trigonometry. Angles on the unit circle are often measured in radians . Radian measure is based on the circumference of the unit circle, C = 2πr . The radius is 1, so 2πr = 2 π. An angle that rotates all the way around the circle is π radians, half-way around is π radians, one-third the way is Trigonometry radians, and so on. The relationship 2π radians = 360° gives us two equations.


These equations help us to convert radian measure to degrees and degree measure to radians. We can convert radians to degrees by multiplying the angle by 180/ π . We can convert degrees to radians by multiplying the angle by π /180.


  • Convert 4π /5 radians to degree measure.
  • Because we are going from radians to degrees, we will multiply the angle by 180/ π .

Trigonometry Examples

  • Convert 5π /6 radians to degree measure.

Trigonometry Examples

  • Convert 48° to radian measure.
  • Because we are going from degrees to radians, we will multiply the angle by π /180.

Trigonometry Examples

  • Convert −72° to radian measure.

Trigonometry Examples

Practice problems for this concept can be found at: Trigonometry Practice Test.

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