= Chord length
{title2=$L=2r\sin(\theta/2)$}
For a circle of radius $r$, a chord with minor central angle $0\leq\theta\leq\pi$ has length
$$
L=2r\sin(\theta/2).
$$
Bisecting the isosceles triangle formed by its two radii gives a right triangle with hypotenuse $r$ and opposite side $L/2$, proving the formula.
If the perpendicular distance from the center to the chord is $d$, the same right triangle gives $L=2\sqrt{r^2-d^2}$. For $r=5,d=3$, the length is eight. The maximum chord is the diameter $2r$, obtained at $d=0$ or $\theta=\pi$.
The corresponding minor arc has length $r\theta$ when the angle is measured in radians. For small angles the chord is approximately that arc length, since $\sin(\theta/2)\sim\theta/2$, but for a nonzero angle the straight chord is strictly shorter than the arc.
Back to article page
