Converting 120 Degrees to Radians: How-To & Tutorial

Instructor: Sharon Linde
Changing from degrees to radians can be confusing. This lesson defines radian, how to convert 120 degrees into radians, and also gives you other common angles measured in radians.

Steps to Solving the Problem

How do we convert 120 degrees to an angle measured in radians?

Degrees and radians are both units of measuring angles. One radian is defined as the angle created at the center of a circle by taking an arc equal to the radius and stretching it along the outside of that circle.

For angles that cover exactly half of the circle, there are 180 degrees and pi radians. Most of you will recall that the first three digits of pi are 3.14.

A graphical depiction showing a single radian
radian

Now that we know what both degrees and radians are, lets get to working the problem.

  • Step 1 - Write our initial equation

angle in radians = angle in degrees x (conversion factor of degrees to radians)

Or

angle in radians = 120 degrees x (conversion factor of degrees to radians)

A conversion factor lets you change one unit of measurement (degrees) to another unit of measurement (radians). We don't yet know what the conversion factor is, but once we do, we'll be able to convert from degrees to radians.

  • Step 2 - Set up the conversion factor

Since we know that an angle of pi radians is the same as an angle of of 180 degrees, we know the conversion factor is:

(pi radians / 180 degrees)

Note that the units we are changing to is on the top of this fraction, and the units we are changing from is on the bottom.

  • Step 3 - Substitute what we know into the equation

We just figured out what the conversion factor is, and we also know that the angle we want to convert is 120 degrees, so we can substitute both of those pieces of information into the equation:

angle in radians = 120 degrees x (pi radians / 180 degrees)

  • Step 3 - Perform the math

Doing the math operation on our equation results in:

angle in radians = (120 x pi) / 180 radians

Since there were units of degrees on both the top and the bottom of the fraction, and since a degree divided by a degree is just 1 without any units we are left with just the units of radians - which is what we wanted.

Solution

We can further simplify our answer by noticing that 120 and 180 are both evenly divisible by 60. Doing this and rearranging slightly we obtain our final answer:

angle in radians = pi x (2/3)

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