Math

The maximum possible distance between their endpoints is the diameter of the circle, namely 2r.

 !

Let the circle have radius rrr.
A chord of length rrr subtends a central angle θ\thetaθ with

r=2rsin⁡θ2⇒1=2sin⁡θ2⇒sin⁡θ2=12,r = 2r\sin\frac{\theta}{2}\quad\Rightarrow\quad 1=2\sin\frac{\theta}{2}\Rightarrow \sin\frac{\theta}{2}=\tfrac12,r=2rsin2θ​⇒1=2sin2θ​⇒sin2θ​=21​,

so θ2=30∘\frac{\theta}{2}=30^\circ2θ​=30∘ and θ=60∘\theta=60^\circθ=60∘.
Thus each chord joins two points on the circle separated by 60∘60^\circ60∘.

The largest possible distance between any two points on the circle is the diameter, 2r2r2r. We can realize that maximum by placing the two chords so that one endpoint of the first chord is diametrically opposite (antipodal to) one endpoint of the second chord. For example, take chord A with endpoints at angles 0∘0^\circ0∘ and 60∘60^\circ60∘ and chord B with endpoints at 180∘180^\circ180∘ and 240∘240^\circ240∘; each chord subtends 60∘60^\circ60∘, and the points at 0∘0^\circ0∘ and 180∘180^\circ180∘ are antipodal, distance 2r2r2r apart.

Therefore the maximum possible distance between their endpoints is 2r\boxed{2r}2r​.