October 28, 2022

Radians to Degrees - Conversion, Formula, Examples

Radians and degrees conversion is a very crucial skill for advanced arithmetic learners to understand.

First, we need to specify what radians are in order that you can understand how this formula is used in practice. After that we’ll take it one step further by looking at a few examples of converting from radians to degrees quickly!

What Is a Radian?

Radians are units of measurement for angles. It is derived from the Latin word "radix," which means ray or nostril, and is a fundamental theory in geometry and mathematics.

A radian is the SI (standard international) unit of measurement for angles, even though a degree is a more generally utilized unit in arithmetic.

That being said, radians and degrees are simply two different units of measure utilized for measuring the same thing: angles.

Note: a radian is not to be mixed with a radius. They are two completely different things. A radius is the distance from the middle of a circle to the border, while a radian is a unit of measure for angles.

Correlation Between Radian and Degrees

We have two ways to think regarding this question. The first method is to figure out about how many radians are present in a full circle. A full circle is equivalent to 360 degrees or two pi radians (exactly). Hence, we can state:

2π radians = 360 degrees

Or simply:

π radians = 180 degrees

The second way to figure out regarding this question is to think about how many degrees exists in a radian. We all know that there are 360 degrees in a full circle, and we also understand that there are two pi radians in a whole circle.

If we divide each side by π radians, we’ll get that 1 radian is approximately 57.296 degrees.

π radiansπ radians = 180 degreesπ radians = 57.296 degrees

Both of these conversion factors are useful relying upon which you're trying to do.

How to Convert Radians to Degrees?

Now that we've went through what radians and degrees are, let's practice how to turn them!

The Formula for Changing Radians to Degrees

Proportions are a useful tool for changing a radian value into degrees.

π radiansx radians = 180 degreesy degrees

Simply plug in your known values to obtain your unknown values. For example, if you wanted to convert .7854 radians to degrees, your proportion will be:

π radians.7854 radians = 180 degreesz degrees

To find out the value of z, multiply 180 with .7854 and divide by 3.14 (pi): 45 degrees.

This formula works both ways. Let’s double-check our workings by changing 45 degrees back to radians.

π radiansy radians = 180 degrees45 degrees

To find out the value of y, multiply 45 with 3.14 (pi) and divide by 180: .785 radians.

Since we've changed one type, it will always work out with another straightforward calculation. In this scenario, after changing .785 from its first form back again, ensuing these steps made exactly what was anticipated -45°.

The formulas work out like this:

Degrees = (180 * z radians) / π

Radians = (π * z degrees) / 180

Examples of Converting Radians to Degrees

Let's attempt a few examples, so these concepts become easier to digest.

Now, we will convert pi/12 rad into degrees. Just like previously, we will place this number into the radians slot of the formula and solve it like this:

Degrees = (180 * (π/12)) / π

Now, let divide and multiply as you normally would:

Degrees = (180 * (π/12)) / π = 15 degrees.

There you have the result! pi/12 radians equivalents 15 degrees.

Let's try one more general conversion and transform 1.047 rad to degrees. Once again, utilize the formula to get started:

Degrees = (180 * 1.047) / π

One more time, you multiply and divide as fitting, and you will wind up with 60 degrees! (59.988 degrees to be exact).

Right away, what happens if you are required to transform degrees to radians?

By using the very exact formula, you can do the converse in a pinch by solving it considering radians as the unknown.

For example, if you wish to change 60 degrees to radians, plug in the knowns and work out with the unknowns:

60 degrees = (180 * z radians) / π

(60 * π)/180 = 1.047 radians

If you remember the equation to find radians, you will get identical answer:

Radians = (π * z degrees) / 180

Radians = (π * 60 degrees) / 180

And there you have it! These are just handful of examples of how to convert radians to degrees and the other way around. Remember the equation and try solving for yourself the next time you are required to make a transformation among radians and degrees.

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