October 28, 2022

Radians to Degrees - Conversion, Formula, Examples

Radians and degrees conversion is a very important skill for advanced mathematics students to understand.

First, we need to define what radians are thereby you can perceive how this formula is used in practice. Thereafter we’ll take a further step by showing a few examples of going from radians to degrees quickly!

What Is a Radian?

Radians are measurement units for angles. It comes from the Latin word "radix," which suggests ray or nostril, and is a critical 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 mathematics.

Simply put, radians and degrees are just two distinct units of measure employed for measuring the same thing: angles.

Note: a radian is not to be confused with a radius. They are two entirely separate things. A radius is the distance from the center of a circle to the perimeter, though a radian is a unit of measure for angles.

Association Between Radian and Degrees

There are two manners to go about regarding this question. The first method is to contemplate about how many radians exists in a full circle. A full circle is equals to 360 degrees or two pi radians (exactly). Therefore, we can state:

2π radians = 360 degrees

Or easily:

π radians = 180 degrees

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

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

π radiansπ radians = 180 degreesπ radians = 57.296 degrees

Both of these conversion factors are helpful depending upon which you're trying to do.

How to Convert Radians to Degrees?

Now that we've covered what radians and degrees are, let's practice how to change them!

The Formula for Giong From Radians to Degrees

Proportions are a useful tool for converting a radian value to degrees.

π radiansx radians = 180 degreesy degrees

With ease plug in your known values to obtain your unknown values. For example, if you wished to turn .7854 radians to degrees, your proportion would be:

π radians.7854 radians = 180 degreesz degrees

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

This formula implied both ways. Let’s verify our workings by changing 45 degrees back to radians.

π radiansy radians = 180 degrees45 degrees

To solve for y, multiply 45 with 3.14 (pi) and divide by 180: .785 radians.

Since we've transformed one type, it will always work with another straightforward calculation. In this scenario, afterwards changing .785 from its original form back again, after these steps created precisely what was anticipated -45°.

The formulas plays out like this:

Degrees = (180 * z radians) / π

Radians = (π * z degrees) / 180

Examples of Going From Radians to Degrees

Let's try some examples, so these theorems become easier to digest.

Now, we will transform pi/12 rad into degrees. Much like previously, we will place this number in the radians slot of the formula and work it out it like this:

Degrees = (180 * (π/12)) / π

Now, let divide and multiply as you normally do:

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

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

Let's try one more common conversion and transform 1.047 rad to degrees. Yet again, use the formula to get started:

Degrees = (180 * 1.047) / π

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

Right away, what to do if you are required to convert degrees to radians?

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

For example, if you have 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 memorized the equation to find radians, you will get the same thing:

Radians = (π * z degrees) / 180

Radians = (π * 60 degrees) / 180

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

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