Lesson Video: Conversion between Radians and Degrees Mathematics

Video Transcript

In this video, we will learn how to convert between radians and degrees and vice versa. Let’s begin by thinking about what a radian is.

Just like we have with degrees, radians are a unit of measure for angles. One radian is defined as the angle made at the center of a circle by an arc whose length is equal to the radius of the circle. Let’s see what that would look like. Let’s say that this is our circle with a center labeled as 𝑂 and here’s the radius of our circle, which we can label as 𝑟. Let’s say we draw another length of the same length of the radius 𝑟 and then we fold it so that it lies along the circle. So then, it looks like this. And we know that the length on the other side of this segment will also be of length 𝑟. The angle that’s created in this segment is one radian. And the question that we’re aiming to answer in this video is to see if we can work out what this angle in radians will be in degrees.

Let’s remember another fact that we should recall about circles. The distance all the way round the outside of this circle is called the circumference. And the circumference of a circle is found by multiplying two times 𝜋 times the radius. A sensible question to ask might be, how many of these arc lengths of 𝑟 would fit in the circumference? And the answer is that two 𝜋 of these arcs would fit on the circumference. We can check this out by remembering that 𝜋 is approximately equal to 3.14. This means that two 𝜋 will be approximately equal to 6.28.

So, if we draw in another arc length of 𝑟, it would look like this, another like this, until we can see that we have six arc lengths plus this little portion remaining. So, if we think that this whole angle at the center of this circle is two 𝜋 radians, then we might think another fact that we know about the angles created in a circle. And that is that the angles created in a circle or by the point are 360 degrees. And we can, therefore, say that two 𝜋 radians is equal to 360 degrees. Knowing this fact will help us to convert between radians and degrees and vice versa.

Notice that if we divide both sides of this conversion by two, we’d get the fact that 𝜋 radians is equal to 180 degrees. This can be very helpful because we know that there is 180 degrees in a semicircle and also on a straight line and even in a triangle. If we wanted, we could even divide that second equation by two on both sides to give us that 𝜋 over two radians is equal to 90 degrees, which is, of course, a right angle. Knowing these three different conversions can be helpful to help us switch between radians and degrees. But we really only need to remember one of them as knowing one of them would allow us to derive the other two.

Let’s now have a look at some questions. And in our first question, we’ll change several angles in degrees into angles in radians.

Convert the following angle measures from degrees to radians. Give your answers in terms of 𝜋 in their simplest form. 90 degrees, 30 degrees, 55 degrees.

In this question, we’re given three different angles in degrees, and we need to change these into a different unit of measurement for angles, radians. One of the key conversions that we can remember between degrees and radians is that 180 degrees is equal to 𝜋 radians. Let’s see how that will help us answer the question of how many radians there are in 90 degrees. Well, we can notice that to go from 180 degrees to 90 degrees, we must divide by two. And so, our radian measurement must also be divided by two. 𝜋 divided by two can be written as 𝜋 over two. And so, the answer is 𝜋 over two radians.

Notice that this answer is given in terms of 𝜋, and it is in its simplest form. Angles in radians are usually given in this form, but we could, of course, use a calculator to convert it to a decimal if required.

Let’s have a look now at converting our 30-degree angle. We can use the same conversion that 180 degrees is equal to 𝜋 radians. And this time, we observe that if we go from 180 degrees to 30 degrees, that’s the same as dividing by six. On the right-hand side, then, we need to also divide by six. And so, we can give our answer that 30 degrees is equal to 𝜋 over six radians. Once again, that’s in terms of 𝜋 and in its simplest form.

So, let’s have a look at this final part. This time, we’re trying to convert 55 degrees into radians. We might already realize that this isn’t going to be quite so simple as 55 degrees isn’t a factor of 180 degrees. Instead of going directly from 180 degrees to 55 degrees, we’ll find it more helpful to find the step in between. We might find it most helpful to think of how we would find one degree, but there are other methods. For example, we could find how many radians there are in five degrees.

But let’s say we use one degree. This time, then, to go from 180 to one degree, we must have divided by 180. Dividing our value in radians by 180 would give us 𝜋 over 180 radians. We now need to think about how we go from one degree to 55 degrees. And that would be by multiplying by 55. So, we’ll need to do that to both sides of this equation. Rather than leaving our answer as 55𝜋 over 180, we can take out a common factor of five. Therefore, we can give our answer that 55 degrees is equal to 11𝜋 over 36 radians. And therefore, we’ve got all of these angles in degrees converted into radians.

In the following question, we’ll see how we can change an angle in radians into an angle in degrees.

Convert 𝜋 over three radians to degrees.

We should remember that, just like degrees, radians are a unit of measurement of angles. We say that one radian is the angle made at the center of a circle by an arc whose length is equal to the radius of the circle. However, in order to actually convert between radians and degrees, we need to remember a key conversion. We might remember either that two 𝜋 radians equals 360 degrees or that 𝜋 radians equals 180 degrees. Recalling either one of these would allow us to convert our value of 𝜋 over three radians.

Let’s say we use the conversion that 𝜋 radians is equal to 180 degrees. We then need to ask ourselves how we go from 𝜋 to 𝜋 over three. Well, we could do that by dividing by three. Therefore, we’ll need to take the value of 180 degrees and divide it by three as well, which gives us 60 degrees. And that’s our answer in degrees for 𝜋 over three radians.

We’ll now have a look at a question where we need to convert between radians and degrees in the context of a problem.

Find the value of two angles in degrees given their sum is 74 degrees and their difference is 𝜋 over six radians. Give your answer to the nearest degree.

In this question, we’re told that there are two angles. We’re also told that their sum is 74 degrees and their difference is 𝜋 over six radians. When we’re answering a question like this, we’ll have to employ different mathematical skills. We’ll need to use a little bit of algebra to solve this problem. And we’ll also need to know how to convert between angles in degrees and angles in radians.

Let’s begin by saying that we can say that our two angles are called 𝑥 and 𝑦. As we’re told that their sum is 74 degrees, we can say that 𝑥 plus 𝑦 is equal to 74 degrees. Next, we’re told that their difference is 𝜋 over six radians. Remember that difference means subtract, so we can write that 𝑥 subtract 𝑦 is equal to 𝜋 over six radians.

Now that we have two equations with two unknowns, we could solve these. However, the problem is that one of these measurements is in degrees and one of the measurements is in radians. We can either find both of these angles in degrees or both in radians. But if we have a look at the question, we need to give our final answer in degrees, so it would make sense to make sure that they’re both in degrees.

Let’s take this angle then of 𝜋 over six radians and write it as a value in degrees. In order to do this, we need to remember an important conversion between radians and degrees. 𝜋 radians is equal to 180 degrees. Some people prefer to remember that two 𝜋 radians is equal to 360 degrees. But either one will allow us to convert these angles. So, if we take the fact that 𝜋 radians is equal to 180 degrees and the value that we have of 𝜋 over six radians is six times smaller, then that means that our angle in degrees must also be six times smaller than 180 degrees, which means that it must be 30 degrees.

Now that we know that this value of 𝜋 over six radians is actually 30 degrees, we can say that 𝑥 minus 𝑦 is equal to 30 degrees. We can now solve this system of equations by either substitution or by elimination. If we choose to use an elimination method and we wanted to eliminate the 𝑦-variable, then we could add together the first equation and the second equation. Adding the two 𝑥-values would give us two 𝑥. 𝑦 subtract 𝑦 would give us zero. And 74 degrees plus 30 degrees would give us 104 degrees.

We can then find the value of 𝑥 by dividing both sides of this equation by two. So, 𝑥 is equal to 52 degrees. We then take this value of 𝑥 and plug it into either the first equation or the second equation. Using the first equation, then, with 𝑥 is equal to 52 degrees, we’d have that 52 degrees plus 𝑦 is equal to 74 degrees. Subtracting 52 degrees from both sides would give us that 𝑦 is equal to 22 degrees. We can, therefore, give our answer that the two angles must be 52 degrees and 22 degrees. And as they’re already whole-value answers, then we don’t need to worry about rounding to the nearest degree.

It is, of course, always worthwhile checking that our answer is correct. When we were solving it, we used this equation 𝑥 plus 𝑦 equals 74 degrees, so let’s check that if we subtract our angles, we would get 30 degrees. And if we have 52 degrees subtract 22 degrees, we would indeed get 30 degrees, confirming that our two angles are 52 degrees and 22 degrees.

In the final question, we’ll solve a problem involving angles in radians and angles in degrees in the context of a triangle.

Two angles in a triangle are 55 degrees and seven 𝜋 over 18 radians. Find the value of the third angle giving the answer in radians in terms of 𝜋.

In this question, we’re given that there’s a triangle. One of the angles, we’re told, is 55 degrees. And the other angle is seven 𝜋 over 18 radians. We need to find out the value of the third angle in this triangle. And we need to give the answer in radians.

The first problem we should notice here is that one of the angles is given in degrees and the other one is given in radians. We might be a little confused and think that radians are something that only involves circles. But just remember that radians are like degrees; they’re just a unit of measurement for angles. What we want to do in this question is to make sure that all the angles are given in the same unit of measurement. We could change them all to degrees, or we could change them all to radians. But we should notice that the question asks for the angle in radians at the end, so it might be more sensible to change our value in degrees into a value in radians.

When it comes to converting between degrees and radians, there are two common conversions we can remember, either 180 degrees is equal to 𝜋 radians or 360 degrees is equal to two 𝜋 radians. Remembering just one of these will allow us to convert any measurement in degrees into a measurement in radians. So, let’s take this conversion of 180 degrees is equal to 𝜋 radians and use it to change 55 degrees into a value in radians. If we take this interim step of finding one degree, then we can notice that to go from 180 to one, we must divide by 180.

This means that we need to do the same on the other side with our value in radians. And 𝜋 divided by 180 can be written as 𝜋 over 180 radians. To go from one degree to 55 degrees, we must multiply by 55. We can then simplify this value on the right-hand side. So, we can say that 55 degrees is equal to 11𝜋 over 36 radians. We can notice, of course, that there is another way we could’ve solved this. Instead of finding one degree, we might have converted five degrees into radians. To go from 180 degrees to five degrees, we must have divided by 36. Then, to change five degrees into 55 degrees, we would’ve needed to multiply both sides by 11. Either way would have given us the value of 11𝜋 over 36 radians.

Now that we’ve calculated that this angle of 55 degrees is equivalent to 11𝜋 over 36 radians, let’s see if we can find this third angle in the triangle. We’ll need to remember that the angles in a triangle add up to 180 degrees. If we call this unknown angle in the triangle 𝑥, we could begin to write that the three angles must sum to — Oh dear! They can’t add up to 180 degrees; it needs to be a value in radians. But we already know that 180 degrees is equal to 𝜋 radians. Therefore, these three angles must add to 𝜋.

We’ll now need to do some fraction arithmetic. And we’ll remember that when we’re adding fractions, they need to have the same denominator. In order to write our second fraction with the denominator of 36, we’ll need to multiply the numerator and denominator by two. So, our second fraction will be 14𝜋 over 36. When we’re adding fractions and they’re the same denominator, we add the numerators. So, we’ll have 25𝜋 over 36 plus 𝑥 is equal to 𝜋. In order to find 𝑥, we subtract 25𝜋 over 36 from both sides of this equation. In order to help us solve this, it might be helpful to take out a factor of 𝜋. So, we’ll have 𝑥 is equal to 𝜋 multiplied by one minus 25 over 36.

Using the fact that this whole value of one must be 36 over 36, then we get that 𝑥 is equal to 11𝜋 over 36 radians. And that’s our answer for the third angle in this triangle. And it’s given in radians and in terms of 𝜋.

Let’s now summarize what we’ve learnt in this video. Firstly, we saw that radians and degrees are both units of measure of angles. We saw that one radian is the angle made at the center of a circle by an arc whose length is equal to the radius of the circle. Finally, we can convert between degrees and radians by remembering one of the following: two 𝜋 radians is equal to 360 degrees, 𝜋 radians is equal to 180 degrees, or 𝜋 over two radians is equal to 90 degrees. Using any of these three would allow us to change between degrees and radians and vice versa.