Question Video: Converting Radians to Degrees and Vice-Versa Mathematics

Video Transcript

Convert 𝜋 divided by three to degrees.

The distance around a circle is called the circumference, and it’s equal to two times 𝜋 times 𝑟, where 𝑟 is the radius. The circumference is the distance around the circle, which is equal to two times 𝜋 times 𝑟, where 𝑟 is the radius. And the radius is the distance from the centre to the circle itself.

An angle of a circle can also be measured in units that are based on arc length. And an arc is like a piece of their circumference, the distance around the circle. One radian is the measure of an angle in standard position with the terminal side that intercepts an arc with the same length as the radius of the circle. So since a circumference of a circle is two times 𝜋 times 𝑟, one complete revolution around the circle is equal to two 𝜋 radians.

And since two 𝜋 radians is the distance around the circle, we can think of it as in degrees and there’re 360 degrees for a whole circle. So because two 𝜋 radians is equal to 360 degrees, degree measure and radian measure are related by the following equation. If we would divide both sides by two, 𝜋 radians is equal to 180 degrees. So we can look at this two ways: we can move 𝜋 radians to the right by dividing and we get one equals 180 degrees divided by 𝜋 radians or we can move the 180 over to the left by dividing and we get one equals 𝜋 radians divided by 180.

So depending on what we’re wanting, whether it’s degrees or radians, we pick which one we need to multiply by. So since we want degrees, we want 𝜋 radians to cancel. So since we have 𝜋 divided by three, 𝜋 is on the numerator, so we want 𝜋 to be on the denominator so the 𝜋 radians go away and we’re left with degrees. So we will multiply by 180 degrees over 𝜋 radians. And as we said before, the 𝜋 divided by three is indeed in radians. So the 𝜋 radians cancel and 180 degrees divided by three is equal to 60 degrees.