Circles

When a chord is drawn in a circle, the shortest distance from the centre to the chord is the perpendicular segment that bisects the chord. If the radius is $r$ and the distance from the centre to the chord is $d$, then each half‑chord has length $\sqrt{r^2-d^2}$. Therefore the full chord length is $2\sqrt{r^2-d^2}$. This comes directly from the right‑triangle formed by the radius, the distance $d$, and half the chord. Knowing $r$ and $d$ lets you compute any chord length, and conversely, knowing a chord length lets you find its distance from the centre.