Can the value of $\pi$ be the denominator of an angle in radians?

Question

I understand that usually problems have $\pi$ in the numerator as this can easily convert to degrees with whole numbers, but is it possible for an angle in radians to be a fraction with $\pi$ in the denominator? I believe this is so as I think $\pi$ is just a constant, but I wanted to confirm.

Answer 1

Yes! Any real number between $0$ and $2\pi$ (including $0$, excluding $2\pi$) can be the measure of an angle. Real numbers outside of this range can also be used to describe the measures of angles, but then the measures of angles are not unique – e.g. an angle of measure $3\pi$ is the same as an angle of measure $\pi$.

Answer 2

Radians measure the "distance" traveled if you start at a point on the unit circle and move along the arc determined by the angle. So the full circle (three hundred sixty degrees) is $2\pi$ since the unit circle has circumference $2\pi$. On the other hand, a right angle covers a quarter of the circle, hence $2\pi/4=\pi/2$ radians.

This is the only reason why $\pi$ shows up in the numerator so much--its easiest for us to think about angles that take up a "reasonable" fraction of the whole circumference, like a fourth, a third, a half, etc., which correspond to $\pi/2$, $2\pi/3$, and $\pi$ radians, respectively.

But you can still use radians to measure all the other "weird" angles that cover any arclength you like. So this could be $1/\pi$ radians, $1/e$ radians, $\phi$ (golden ratio) radians, etc...