October 28, 2022

Radians to Degrees - Conversion, Formula, Examples

Radians and degrees conversion is a very essential skill for advanced mathematics students to grasp.

First, we need to specify what radians are so that you can see how this theorem works in practice. Then we’ll take a further step by exhibiting 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 nostril or ray, and is a critical theory in geometry and mathematics.

A radian is the SI (standard international) measuring unit for angles, although a degree is a more commonly utilized unit in arithmetic.

In other words, radians and degrees are simply two different units of measure employed for measuring the exact thing: angles.

Note: a radian is not to be mixed with a radius. They are two absolety distinct things. A radius is the length from the center of a circle to the perimeter, whereas a radian is a unit of measure for angles.

Relationship Between Radian and Degrees

There are two ways to think regarding this question. The first way is to think about how many radians are present in a full circle. A full circle is equals to 360 degrees or two pi radians (exactly). So, we can state:

2π radians = 360 degrees

Or simplified:

π radians = 180 degrees

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

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

π radiansπ radians = 180 degreesπ radians = 57.296 degrees

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

How to Convert Radians to Degrees?

Since we've covered what radians and degrees are, let's practice how to convert them!

The Formula for Converting Radians to Degrees

Proportions are a beneficial tool for changing a radian value to degrees.

π radiansx radians = 180 degreesy degrees

Just put in your given values to derive your unknown values. For example, if you wished to convert .7854 radians into degrees, your proportion will be:

π radians.7854 radians = 180 degreesz degrees

To solve for z, multiply 180 with .7854 and divide by 3.14 (pi): 45 degrees.

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

π radiansy radians = 180 degrees45 degrees

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

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

The formulas solves like this:

Degrees = (180 * z radians) / π

Radians = (π * z degrees) / 180

Examples of Converting Radians to Degrees

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

Now, we will transform 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 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, use the formula to get started:

Degrees = (180 * 1.047) / π

Yet again, you multiply and divide as suitable, and you will find yourself with 60 degrees! (59.988 degrees to be precise).

Now, what to do if you have to transform degrees to radians?

By employing 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 convert 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 solve for radians, you will get identical answer:

Radians = (π * z degrees) / 180

Radians = (π * 60 degrees) / 180

And there you have it! These are just some of the examples of how to transform radians to degrees and the other way around. Remember the formula and try it out for yourself the next time you need to make a transformation between radians and degrees.

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