Length of a Chord

Length of the chord at perpendicular distance d from the centre of a circle of radius r. Maximum (diameter) when d = 0.

Let a chord $AB$ of a circle with centre $O$ and radius $r$ be at perpendicular distance $d$ from the centre. Let $M$ be the foot of the perpendicular from $O$ to $AB$.

Since the perpendicular from the centre bisects the chord, $AM = MB$.

In the right triangle $OMA$:

$$OA^2 = OM^2 + AM^2$$ $$r^2 = d^2 + AM^2$$ $$AM = \sqrt{r^2 - d^2}$$

The full chord length:

$$AB = 2 \cdot AM = 2\sqrt{r^2 - d^2}$$

Boundary cases:

• $d = 0$: chord is a diameter, $\ell = 2r$
• $d = r$: chord degenerates to a point (tangent condition)
• $d > r$: the line does not intersect the circle

This formula is also the basis for finding the perpendicular distance given the chord length.