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

Notice that since degrees is on top and at the bottom, we can cancel it. Notice also that we divided both the numerator and denominator of (30 π radians) / 180 by 30 to simplify it.

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. 

Convert degrees to radians

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.

180 degrees π radians
? x

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)

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