October 28, 2022

Radians to Degrees - Conversion, Formula, Examples

Radians and degrees conversion is a very important ability for advanced mathematics students to grasp.

First, we need to specify what radians are so that you can understand how this formula works in practice. Then we’ll take it one step further by showing a few examples of going from radians to degrees easily !

What Is a Radian?

Radians are measurement units for angles. It is originated from the Latin word "radix," which means ray or nostril, and is a critical idea in geometry and mathematics.

A radian is the SI (standard international) unit of measurement for angles, although a degree is a more frequently used unit in arithmetic.

In other words, radians and degrees are simply two separate 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 edge, though a radian is a measuring unit for angles.

Association Between Radian and Degrees

We have two manners to go about regarding this question. The first method is to think about how many radians there are in a full circle. A full circle is equal to 360 degrees or two pi radians (precisely). Hence, we can say:

2π radians = 360 degrees

Or simply:

π radians = 180 degrees

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

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

π radiansπ radians = 180 degreesπ radians = 57.296 degrees

Both of these conversion factors are helpful relying on what you're trying to get.

How to Change Radians to Degrees?

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

The Formula for Changing Radians to Degrees

Proportions are a beneficial tool for turning a radian value into degrees.

π radiansx radians = 180 degreesy degrees

Simply plug in your given values to obtain your unknown values. For example, if you wished to turn .7854 radians into degrees, your proportion would be:

π radians.7854 radians = 180 degreesz degrees

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

This formula works both ways. Let’s recheck our operation by changing 45 degrees back to radians.

π radiansy radians = 180 degrees45 degrees

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

Since we've converted one type, it will always work out with different simple calculation. In this case, afterwards changing .785 from its first form back again, following these steps made precisely what was predicted -45°.

The formulas plays out like this:

Degrees = (180 * z radians) / π

Radians = (π * z degrees) / 180

Examples of Changing Radians to Degrees

Let's try some examples, so these ideas become simpler to digest.

At the moment, we will convert pi/12 rad to degrees. Just like previously, we will plug this value in the radians slot of the formula and calculate it like this:

Degrees = (180 * (π/12)) / π

Now, let divide and multiply as you usually 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 appropriate, and you will wind up with 60 degrees! (59.988 degrees to be almost exact).

Now, what happens if you want to transform degrees to radians?

By using the very exact formula, you can do the opposite in a pinch by solving for radians as the unknown.

For example, if you wish to transform 60 degrees to radians, put 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 formula to find radians, you will get the same thing:

Radians = (π * z degrees) / 180

Radians = (π * 60 degrees) / 180

And there it is! These are just a few examples of how to convert radians to degrees and the other way around. Bear in mind the equation and try it out for yourself the next time you are required to make a transformation from or to radians and degrees.

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