More About Angle Conversions

Angles refer to a measure of the opening between to rays (or line segments), relative to the opening between segments in a circle, that start from the center of the circle. There are different systems, or conventions to measure that degree of opening. One system is the system of degrees , in which the opening is measured as 0 ^o when the two segments overlap (so there is no opening at all), and 360 ^o represents to the opening of the whole circle. Any other angle is measured in degrees proportionally to the amount of opening between 0 ^o and 360 ^o .

Another system used is radians , which uses a different approach. It measures an angle based on the "number of radiuses" the arc length of the segment in the circle determined by the angle represents. With that into consideration, the angle in radians corresponding to the full circle is \(2\pi\) radians, because the arc length of the full circle is \(2\pi r\), so it is \(2\pi\) times the radius \(r\).

How to Convert Degrees to Radians?

You can use this degrees to radians calculator in the following way: If you have an angle \(d\) in degrees, the angle in radians \(r\) is computed as follows:

\[r = \frac{2\pi d}{360} = \frac{\pi d}{180} \]

How to Convert Radians to Degrees?

You can use this radians to degrees calculator in the following way: If you have an angle \(r\) in radians, the angle in degrees \(d\) is computed as follows:

\[d = \frac{360 r}{2\pi} = \frac{180 r}{\pi} \]

Example: 90 degrees in radians

Taking the above formulas, we are given the degrees (90 degrees), and we want to compute the corresponding radians using the following formula

\[r = \frac{2\pi d}{360} = \frac{90\pi}{180} = \frac{\pi}{2} \]

As we can see, by using the above formula we have a way of expressing degrees to radians, in terms of \(\pi\).