Radians To Degrees


Instructions: Use this radians to degrees calculator with steps to convert an angle in radians to degrees. All you have to do is type the angle in radians, and the calculator will show you how to convert it to degrees, showing all the steps:


Angle in Radians (Ex. '1', '2pi', etc) =


More About Radians to Degrees Conversion

What are radians and why we want to convert them into degrees? In the end, all angle measures are arbitrary, and and are tied to how big an opening of two rays with a common origin opening is. Indeed, angles refer to a measure of the opening between to line segments with respect to the circle

When we measure an angle in degrees, an angle of 360o corresponds to the full opening of a circle. All other angles that are measured in degrees will be proportional to the amount of the opening associated for the angle. For example, if the opening of the angle corresponds to half opening of the circle, the angle will be half of what is for the full opening, this is half of 360o, which is 180o

Another system to measure angles the radians system, which utilizes a different approach, but yet it is the same principle. In this case, the angle depends on the opening, but this opening is measured based on how many radiuses are represented by the arc length of the segment in the circle that is determined by the corresponding angle. We know that the full opening of the circle has a perimeter of \(2\pi\), which otherwise said indicates that we can fit \(2\pi\) radiuses in the full opening of the circle, and this indicates that the full opening of the circle corresponds to \(2\pi\) radians.

How to Convert Radians to Degrees?

So, what is the formula to convert radians to degrees? If you have an angle that is measured as \(r\) in radians, the angle in degrees \(d\) is computed as follows:

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

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

Example 1: Convert \(\pi\) radians into degrees.

Solution: All you need to do is to plug \(r\) into the above formula. In this case, \(r = \pi\) radians, so we get

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

Example 2: Now convert \(\displaystyle \frac{3\pi}{4}\) radians into degrees.

Solution: Same as with the previous example, we need to plug \(r\) into the above formula. In this case, \(\displaystyle r = \frac{3\pi}{4}\) radians, so we get

\[d = \frac{180 r}{\pi} = \frac{180 \cdot \frac{3\pi}{4}}{\pi} = 135^{o} \] so then \(\displaystyle \frac{3\pi}{4}\) radians corresponds to 135o degrees. 135 degrees

For those of you who work with Excel, you can use the function "=DEGREES(r)" to convert an angle in radians to degrees.

If conversely you can to convert degrees of radians, you should use this other calculator.

You can explore other trigonometric calculators, such as our double angle calculator, among many others.




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