Definition: When converting between radians and degrees, it’s important to remember that there are 360­° in a circle, and 2π radians in a circle.

The following formulas can be used to convert between radians and degrees:

Another method for converting between radians and degrees is setting up a proportion to solve for the unknown value.

Example 1: Convert 72° to radians.

Solution

Step 1: Plug the given value into the formula.

= (π/180) * 72
= 72π /180

Step 2: Simplify.

72π/180 = 2π/5

Example 2: Convert -5 radians to degrees.

Step 1: Set up a proportion using x for the unknown degree value. We’ll keep degrees on top and radians on bottom to ensure the ratios are equivalent:

$\dpi{150} \bg_white \frac {x^{\circ}} {-5} = \frac {360^{\circ}} {2 \pi}$

Step 2: Cross multiply and simplify to solve the proportion for x.

2π*x = -1800
πx = -900
x = -900/π = -286.5°