Converting from radians to degrees

To convert from radians to degrees, just multiply by (180 degrees) / (π radians). Let us briefly see why all we need to do is to multiply by (180 degrees) / (π radians).

60 degrees or π/3 radians

Recall that the circumference of a circle is 2πr. Therefore, there are 2π radians in any circle. 

2π radians is equivalent to 360 degrees or 2π radians = 360 degrees.

Divide both sides of 2π radians = 360 degrees by 2.

2π radians / 2 = 360 degrees / 2

π radians = 180 degrees

Now divide both sides of π radians = 180 degrees by π radians.

π radians / π radians = 180 degrees / π radians

1 = 180 degrees / π radians

1 radian = 180 degrees / π radians

Since 1 radian is equal to 180 degrees / π radians, we can just multiply any amount greater than 1 radian by 180 degrees / π radians to get the answer in degrees.

A few examples showing how to convert radians to degrees


Example #1: Convert (π/3) radians to degrees

(π / 3) radians  = (π radians / 3) × 180 degrees / π radians

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

                        = 180 degrees / 3

                        = 60 degrees

Notice that since π radians is on top and at the bottom, we can cancel it.

The figure above shows 60 degrees or π/3 radians.

If you were converting (-π/3) radians to degrees, the answer would be -60 degrees.

Example #2: Convert (π/2) radians to degrees

(π / 2) radians  = (π radians / 2) × 180 degrees / π radians

                        = (π radians × 180 degrees) / 2 × π radians

                        = 180 degrees / 2

                        = 90 degrees

If you were converting (-π/2) radians to degrees, the answer would be -90 degrees.

Example #3: Convert (-π) radians to degrees

First, notice that (-π) radians = (-π / 1) radians

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

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

                          = -180 degrees / 1

                          = -180 degrees

Example #4: Convert (3π/4) radians to degrees

(3π / 4) radians  = (3π radians / 4) × 180 degrees / π radians

                          = (3π radians × 180 degrees) / 4 × π radians

                          = (540 degrees × π radians) / 4 × π radians

                          = 540 degrees / 4

                          = 135 degrees

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