Converting Radians and Degrees

You are probably familiar with degrees as a measure of how large an angle is, but another way of describing angles is with radians. As you approach pre-calculus and your upper years of mathematics, degrees will become less and less frequent as radians become the norm, so it’s a good idea to get used to them early, especially if you plan on studying mathematics.

Degrees work by dividing a circle into 360 equal parts, and radians work the same way, except a circle has 2π radians and π or pi radians equal one-half of the circle or 180 degrees, which is important to remember.

In order to convert angles from degrees to radians, then, students must learn to multiply the measurement of the degrees by pi divided by 180. In the example of 45 degrees in radians, one can simply reduce the equation of r = 45π / 180 to π/4, which is how you would leave the answer to express the value in radians.

Conversely, if you know what an angle is in radians and you want to know what the degrees would be, you multiply the angle by 180/π, and thus 5π radians in degrees will equal 900 degrees—your calculator has a pi button, but in case it's not handy, pi equals 3.14159265.

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How to Determine the Geometry of a Circle

By Deb Russell

Identifying Degrees and Radians

Degrees are units of measurements valued one through 360 that measure the sections or angles of a circle while radians are used to measure the distance traveled by angles. Whereas there are 360 degrees in a circle, each radian of distance moved along the outside of the circle is equal to 57.3 degrees.

Essentially, radians measure the distance traveled along the outside of the circle as opposed to the view of the angle that degree takes up, which simplifies solving problems that deal with measurements of distance traveled by circles like tire wheels.

Degrees are much more useful for defining the interior angles of a circle than for how the circle moves or what distance is traveled by moving along the circle instead of merely looking at it from one perspective while radians are more appropriate for observing natural laws and applying to real-world equations. In either case, they're both units of measurements which express the distance of a circle—it's all a matter of perspective!

The Benefit of Radians Over Degrees

Whereas degrees can measure the internal perspective of angles of the circle, radians measure the actual distance of the circumference of a circle, providing a more accurate assessment of distance traveled than degrees which rely on a 360 scale.

Additionally, in order to calculate the actual length of a segment of a circle with degrees, one must do more advanced computations that include the use of pi to arrive at a product. With radians, the conversion to distance is much easier because a radian views a circle from the perspective of distance rather than the measurement of internal angles alone.

Basically, radians already factor in distance as part of the basis for the equation for defining a radian's size, which makes them more versatile in use than degrees.