Convert Angles between Degrees and Radians

Another way to represent an angle is by using radians. Without getting too technical, a radian is defined as the angle between 2 radii (plural for radius) of a circle where the arc between them has length of one radius.

While it’s not stated in the animation, the arc length equals the length of one radius when the angle is approximately 57.2958°. This leads us to two new conversion ratios:


Extending this notion further:


We can use any of these conversion ratios to convert between radians and degrees. Let’s watch three examples:

Now you might run into a situation where you’ll be expected to convert an angle in radians to degrees, minutes, seconds – which was covered extensively in the previous lesson. To make this conversion, you’ll need the conversion ratios introduced above along with ones stated below:

  • 1 minute (‘) is equal to 1/60 of a degree.
  • 1 second (”) is equal to 1/3600 of a degree.

A video demonstration is shown below:

  • Keep in mind that the answer provided doesn’t take into account significant figures.