Converting 120 Degrees to Radians: How-To & Tutorial

Sharon Linde

Sharon has a Masters of Science in Mathematics

Expert Contributor
Robert Ferdinand

Robert Ferdinand has taught university-level mathematics, statistics and computer science from freshmen to senior level. Robert has a PhD in Applied Mathematics.

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

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)


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.


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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Additional Activities

Practice Problems Converting Degrees to Radians for Angles

Key Terms

  • Radian: Angle measure as the length of the arc that subtends the angle, divided by the length of the radius of the circle.
  • Degree: Traditional angle measure, as measured by a protractor for example.

Materials Needed

  • Paper
  • Pencil
  • Calculator

Example: Convert 60 Degrees to Radians

Solution: Proceed as follows:

Step 1: Since the length of the arc that subtends an angle of 180 degrees in a circle with radius = 1 unit is pi, therefore, 180 degrees = pi radians, giving us a unit converting factor CF as follows:

CF = pi (radians) / 180 (degrees), or simply pi / 180.

Step 2: Multiply the degree measure (60 degrees) of the given angle in the question by the CF in Step 1 above to obtain the equivalent radian measure as follows:

60 Degrees x pi/180 = pi/3 = 1.05.

Answer: pi/3 or 1.05 Radians.

Now, let's see you practice some questions on your own, using the steps outlined above (Never Forget to Show All Your Work):

(a) Convert 30 Degrees to Radians.

(b) Convert 150 Degrees to Radians.

(c) Convert 180 Degrees to Radians.

(d) Convert -40 Degrees to Radians.

Answers (To Check Your Work):

(a) pi/6 or 0.52 Radians.

(b) 5pi/6 or 2.62 Radians.

(c) pi or 3.14 Radians.

(d) -2pi/9 or -0.70 Radians.

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