October 28, 2022

Radians to Degrees - Conversion, Formula, Examples

Radians and degrees conversion is a very essential ability for advanced math learners to grasp.

First, we need to specify what radians are so that you can see how this formula is used in practice. Then we’ll take it one step further by showing some examples of going from radians to degrees with ease!

What Is a Radian?

Radians are units of measurement for angles. It is derived from the Latin word "radix," which implies ray or nostril, 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 used unit in arithmetic.

In other words, radians and degrees are merely two distinct units of measure used for measuring the exact thing: angles.

Note: a radian is not to be mistaken with a radius. They are two absolety distinct concepts. A radius is the length from the center of a circle to the edge, though a radian is a measuring unit for angles.

Relationship Between Radian and Degrees

We have two manners to think about this question. The first way is to contemplate about how many radians there are in a full circle. A full circle is equals to 360 degrees or two pi radians (precisely). So, we can say:

2π radians = 360 degrees

Or simplified:

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 know that there are 360 degrees in a full 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 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 Change Radians to Degrees?

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

The Formula for Giong From 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 known values to derive your unknown values. For example, if you wished to convert .7854 radians to degrees, your proportion would 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 works both ways. Let’s double-check our work by reverting 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 out with another simple calculation. In this instance, afterwards changing .785 from its original form back again, ensuing these steps produced perfectly what was expected -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 go through a handful of examples, so these theorems become easier to digest.

At the moment, we will convert pi/12 rad into degrees. Just like previously, we will put this value 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 do:

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

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

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

Degrees = (180 * 1.047) / π

Once again, you multiply and divide as appropriate, and you will end up with 60 degrees! (59.988 degrees to be precise).

Right away, what happens if you have to change degrees to radians?

By employing the very exact formula, you can do the contrary in a pinch by work out with 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 remember the formula to solve for 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 transform radians to degrees and the other way around. Bear in mind the formula and try it out for yourself the next time you need to make a transformation among radians and degrees.

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