Question

In a unit circle, what is the length of an arc that subtends an angle of π/4 radians?

Answer 1

A unit circle has radius 1, and thus circumference $2\pi$.

Since an angle of $\frac{\pi}{4}$ is one eighth of a whole turn, the length of an arc that subtends an angle of  $\frac{\pi}{4}$ radians will be one eighth of the whole circumference:

$l = \dfrac{2\pi}{8} = \frac{\pi}{4}$

In fact, the radians have the property that, in the unit circle, the length of the arc is exactly the measure of the angle. In general, you have

$l = r\cdot\alpha$

where l is the length of the arc, r is the radius and $\alpha$ is the angle in radians. So, if $r=1$, you have $l=\alpha$