Angles can be measured in degrees or radians. This lesson will explore the difference and provide you with a simple calculation that can be used to convert degrees into radians and radians into degrees.

Converting Between Radians and Degrees

If you ever travel to another country, the first thing you’ll likely notice is that people are usually speaking a different language. You might have a little luck understanding some things, but you’ll quickly find out that even when you are trying to say something simple, it can be very frustrating.

In math, this same thing can happen by using different units of measure. By using different units of measure, a number can mean completely different things.

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When it comes to angles, there are multiple ways to measure size, but the two most common units of measure are degrees and radians. You might already be familiar with one or the other, but knowing how to switch back and forth can be quite helpful as well.

The Translation

If you were trying to speak another language, you might find a translation book helpful. It would show you the words you already know in your own language and will also show you how to say the same words in a different language. Trying to go from degrees to radians and vice versa is the same. We need something that compares the two so we can use it to go back and forth.

In this case, it is not a book, but rather a special number that we can multiply by. To understand where this number comes from, let’s look at the unit circle.

In the unit circle, or any circle for that matter, there are 360 degrees that make up the whole circle. In radians, we say there are 2 pi radians that make up the whole circle. We will use these two key numbers to set up a proportion, which is a fancy way to say that we will make the key numbers into a fraction and use that fraction to convert the measurements. When converting degrees to radians or radians to degrees, the key numbers that we use for the proportion are 2 pi and 360 degrees.

The last step in understanding the proportion is to know where the numbers go. The answer is; it depends. Don’t worry; I won’t just leave you hanging. There’s a simple trick that will help you place the numbers correctly. What you want goes on top. Remember this by thinking that the measurement you want is the top priority, and the measurement you already have is the lowest priority.

Let’s look at some examples.

From Degrees to Radians

First, we will look at translating degree measures into radian measures.

Let’s say we were given a measure in degrees and asked for the radian value. Remember, the two components of our proportion are 360 degrees and 2 pi radians. What we want is the radian measure; it is our top priority. So, we put 2 pi on the top of the fraction. What we already have is the degree measure, so we put 360 degrees on the bottom of the fraction.

Now, we take our degree measure and multiply. If we were given 45 degrees and asked to convert to radians, it would look like this:
45 degrees multiplied by our proportion, which is 2 pi/360 degrees.

Before we jump straight into the calculation, it’s important that we see what happens with our units of measure. We have 45 degrees on top. We also have 2 pi radians on the top of the fraction. On the bottom, we only have degrees. Since we have the measure of degrees on the top and bottom, they cancel each other out.

Now that we have (45 * 2 pi) / 360, we can begin to calculate.
45 * 2 = 90. This leaves us with 90 pi radians / 360.
90 / 360 = 1 / 4. We are left with (1 * pi) / 4.
The 1 is understood, so we can simplify and leave it out. Our answer is pi/4 radians.
We know that a 45 degree angle and a pi/4 radians angle are the same.

From Radians to Degrees

Now, let’s take a look at the whole process in reverse.

Let’s say we were given an angle measure of 5pi/6 radians and were asked to convert it to degrees. We always need to begin by setting up our proportion. Our two numbers are 360 degrees and 2 pi radians. In this case what we want is degrees, so getting degrees is our top priority. You would place 360 degrees on the top part of the fraction. Radians is what we already have, so it is the low priority. 2 pi radians goes on the bottom.

Now that we have our proportion set up, let’s multiply.
We have 5 pi/6 radians * 360 degrees / 2 pi radians.
We multiply these and get (5 pi radians * 360 degrees) / (6 * 2 pi radians).

Before we compute the numbers, let’s look at the units of measure. We have both radians and degrees on top, but we divided by radians. Since we have radians on top and bottom, they cancel each other out. Now we can move on to the numbers.

At this point, we could just type them into a calculator, but we can do some quick simplification steps and make everything easier. First we notice that there is a pi on the top and bottom of the fraction. They cancel each other out, and we don’t have to worry about them.

We have this large number of 360 in the numerator, and we can probably factor it with terms in the denominator. We can see that 360 is an even number, so we know it is divisible by the 2 in the denominator. Let’s do that.
This gives us 180.
180 / 6 = 30
This leaves us with 5 * 30 degrees.
5 * 30 degrees leaves us with our answer: 150 degrees.
We know that 5 pi/6 radians and 150 degrees are the same.

Lesson Summary

Let’s review. If you need to translate degrees to radians or radians to degrees, you need to set up a proportion that will translate for you. There are two numbers that we use to make the proportion, and they are directly from the full unit circle. A full circle is 360 degrees or 2 pi radians.

Now that we know these two numbers, we need to figure out what measure we want; it is our top priority. It goes on the top of the fraction. The measure we already have is our low priority, so it goes on the bottom.

Then, we cancel out unnecessary units and do the math.

There you have it. When you’re working with angles and aren’t speaking the right language, don’t worry. Just remember to use your translator. Hasta luego!

Learning Outcome

After viewing the video lesson, you should be able to calculate for degrees to radians or radians to degrees by always keeping your top priority in mind.