Let $r$ be a natural number, and consider the $\mathbb{Z}^2$ lattice points $S$ inside or on the circle $C$ of radius $r$ centered on the origin. Let $P$ be the convex hull of $S$; so $P$ is inscribed in $C$. I would like to "triangulate" $P$ in a special sense.

A *triangulation* of $P$ is a partition of $P$ into triangles whose
interiors are pairwise disjoint, whose corners are points of $S$,
and such that every point of $S$ is on the boundary of a triangle—either
a corner or on the interior of an edge.

Define the *weight* of a triangulation of $P$ as the sum of the
Euclidean lengths of the segments comprising the triangulation.
(Each segment is counted once even if shared between two triangles.)

. What is the minimum weight triangulation of $P$ as a function of $r$?Q

For $r=1$, the shortest (minimum weight) triangulation has length $2 + 4 \sqrt{2} \approx 7.66$—a split diamond. But even for $r=2$, the minimum length is not obvious (to me). Here are four different triangulations for $r=2$ (where $S$ is a subset of a $5 \times 5$ grid) and their associated lengths. (Pardon any calculation errors.)

^{ $r=2$: $8+12 \sqrt{2} \approx 24.97$. }

Below is just one triangulation for $r=3$, where $S$ is a subset of a $7 \times 7$ grid:

^{ $r=3$: $26+4 \sqrt{2}+8 \sqrt{5}+4 \sqrt{10} \approx 62.19$. }

I am not seeing an obvious pattern. Has this been investigated in the literature? Are there at least asymptotic bounds?

**Update**(

*10Sep2018*). Here are @WlodekKuperberg's shorter triangulations for $r=2,3$:

**Update** (*11Sep2018*).
Here is one triangulation for $r=4$, in a $9 \times 9$ grid:

^{ $r=4$: $44+12 \sqrt{2}+8 \sqrt{5}+4 \sqrt{17} \approx 95.35$. }