Calculators Geometry

Degrees to Radians

Instructions: Use this degrees to radians calculator that shows all the steps to convert an angle in degrees to radians. Please type the desired angle in degrees, and the calculator will show you how to convert it to radians, showing all the steps:

More About this Degrees to Radians Calculator

Why do we have angles measured in degrees and angles measured in radians? The answer is simple: because different angle measuring systems use different references to denominate the measure of the full circle.

Different references for the full circle

When we measure an angle in degrees, the full opening of a circle will correspond to an angle of 360^o. Then, the angles are computed proportionally to that references. This is, an angle corresponding to have of the full circle opening will be half of what is for the full opening, this is half of 360^o, which is 180^o

Now, if we measure an angle in radians the angle corresponding to the full opening of the circle is set as the perimeter of a circle of radius 1, this is \(2\pi\). In other words, you are measuring an angle in terms of how many radiuses need to be used to described the opening. This way, a 360 ^o angle corresponds to \(2\pi\) radians.

How to Convert Degrees to Radians?

Now, if you want to actually go and transform an angle in degrees to radians, what do you? This is what you do: If you have an angle \(d\) measured in degrees, and you want to convert it in a n equivalent angle \(r\) that is measured as in radians, you use the following formula:

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

The above formula applies whether you are converting decimals or fraction degrees to radians. All you need to do is to plug the number of degrees \(d\) into the formula

Example 1: Convert 45 ^o into radians.

Solution: All you need to do is to plug \(d\) into the above formula. In this case, \(d = 45^o\), so we get

\[r = \frac{\pi d}{180} \pi \frac{ 45}{180} = \frac{\pi}{4} \] so then 180 ^o degrees corresponds to \(\frac{\pi}{4}\) radians.