October 28, 2022

Radians to Degrees - Conversion, Formula, Examples

Radians and degrees conversion is a very crucial skill for advanced math students to understand.

Initially, we need to explain what radians are so that you can understand how this theorem is used in practice. After that we’ll take it one step further by looking at a few examples of changing 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 suggests ray or nostril, and is a fundamental idea in mathematics and geometry.

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

In other words, radians and degrees are just two separate units of measure used for measuring the identical thing: angles.

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

Association Between Radian and Degrees

We have two manners to think about this question. The first way 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 (precisely). So, we can say:

2π radians = 360 degrees

Or simplified:

π radians = 180 degrees

The next way to figure out about this question is to think about how many degrees are present in a radian. We understand that there are 360 degrees in a complete circle, and we also recognize that there are two pi radians in a full 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 beneficial depending upon which you're trying to do.

How to Go From Radians to Degrees?

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

The Formula for Converting Radians to Degrees

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

π radiansx radians = 180 degreesy degrees

With ease put in your given values to derive your unknown values. For instance, if you are required to turn .7854 radians to degrees, your proportion will be:

π radians.7854 radians = 180 degreesz degrees

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

This formula can be implemented both ways. Let’s double-check our operation by reverting 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.

Once we've transformed one type, it will always work out with different simple calculation. In this case, afterwards changing .785 from its first form back again, ensuing these steps made perfectly what was anticipated -45°.

The formulas solves like this:

Degrees = (180 * z radians) / π

Radians = (π * z degrees) / 180

Examples of Going From Radians to Degrees

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

At the moment, we will change pi/12 rad to degrees. Much like previously, we will plug 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 generally would:

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

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

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

Degrees = (180 * 1.047) / π

Once again, you multiply and divide as appropriate, and you will find yourself with 60 degrees! (59.988 degrees to be almost exact).

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

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

For example, if you want to transform 60 degrees to radians, plug in the knowns and solve for 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 change radians to degrees and the other way around. Remember the equation and see for yourself the next time you need to make a transformation among radians and degrees.

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