Video: Converting Radians to Degrees

Video Transcript

Convert the following angle measures from radians to degrees. Give your answers correct to the nearest degree where necessary. 𝜋 over three radians, 𝜋 over seven radians, and three 𝜋 over eight radians.

Let’s recall what we already know about radians. One full turn is equal to both 360 degrees and two 𝜋 radians. Halving this gives us 𝜋 radians equals 180 degrees. Let’s take this relationship further. Dividing both sides by 𝜋 will give us the value of one radian. 180 over 𝜋 degrees is therefore equal to one radian. So to change radians into degrees, we can simply multiply by 180 over 𝜋.

Let’s see how this helps us. Multiplying by 180 over 𝜋 in our first example gives us 𝜋 over three times 180 over 𝜋. Did you notice that there’s a common factor of 𝜋? We can therefore cancel this through, leaving us with a third times 180 over one. Multiplying as normal gives us 180 over three or 60 degrees. 𝜋 over three radians is therefore equal to 60 degrees.

Now, consider our second example. Once again, we can multiply by 180 over 𝜋 to convert our radians into degrees. Just like in the last example, we have a common factor of 𝜋. We can therefore divide through by 𝜋 to leave us with a seventh times 180 over one. Multiplying here gives us 180 over seven, which doesn’t give us such a nice number. However, the question told us to give our answers correct to the nearest degree. 180 divided by seven is 25.714 and so on. This gives us 26 degrees correct to the nearest degree. 𝜋 over seven radians is therefore equal to 26 degrees correct to the nearest degree.

At this point, you may wish to pause the video and give the final example a go yourself. Just like in the two previous examples, we are gonna start by multiplying by 180 over 𝜋. Once again, we can divide through by this common factor of 𝜋. This leaves us with three-eighths times 180 over one. Three-eighths times 180 over one is equal to 67.5. Correct to the nearest degree, this is 68 degrees. Three 𝜋 over eight radians is therefore equal to 68 degrees correct to the nearest degree.