How to convert  from degrees to radians

To convert from degrees to radians, just multiply by (π  radians) / (180 degrees). The math is shown below.

Examples showing how to convert from degrees to radians

Example #1: Convert 30 degrees to radians

30 degrees    = 30 degrees × π radians / 180 degrees

                      = (30 degrees / 1) × (π radians / 180 degrees)

                      = (30 degrees × π radians) / 1 × 180 degrees

                      = (30 π radians) / 180 

                      = π radians / 6

Example #2: Convert -90 degrees to radians

-90 degrees    = -90 degrees × π radians / 180 degrees

                      = (-90 degrees / 1) × (π radians / 180 degrees)

                      = (-90 degrees × π radians) / 1 × 180 degrees

                      = (-90 π radians) / 180 

                      = -π radians / 2

Example #3: Convert 120 degrees to radians

120 degrees    = 120 degrees × π radians / 180 degrees

                        = (120 degrees / 1) × (π radians / 180 degrees)

                        = (120 degrees × π radians) / 1 × 180 degrees

                        = (120 π radians) / 180 

                        = 2π radians / 3

Did you understand the examples above clearly? Did you not understand them at all? The figure below summarizes the concept. 

Why did we multiply by (π  radians) / (180 degrees) when converting from degrees to radians?

We know that 360 degrees = 2π radians

Just divide both sides of the equation 360 degrees = 2π radians by 2.

(360 degrees)/2 = (2π radians)/2

180 degrees = π radians

Now let ? represent the amount in degrees that you would like to convert to radians and x the converted amount in radians. You can set up first an "if-then" sentence.

"If" 180  degrees equal  π radians,

"then" ?  is equal to x .

Then, you can pull out all the numbers in red and put them in a table in the same order they are listed above.

The proportion is 180 degrees / ? = π radians / x

After cross multiplication, we get 180 degrees × x = ? × π radians

x = (? × π radians) / 180 degrees

x = ? × (π radians / 180 degrees)

Therefore, if ? is the amount in degrees, to convert from degrees to radians, just multiply by (π radians / 180 degrees)