We study the minimum total weight of a disk triangulation using vertices out of
{
1
,
…
,
n
}
\{1,\ldots ,n\}
, where the boundary is the triangle
(
123
)
(123)
and the
(
n
3
)
\binom {n}3
triangles have independent weights, e.g.
E
x
p
(
1
)
\mathrm {Exp}(1)
or
U
(
0
,
1
)
\mathrm {U}(0,1)
. We show that for explicit constants
c
1
,
c
2
>
0
c_1,c_2>0
, this minimum is
c
1
log
n
n
+
c
2
log
log
n
n
+
Y
n
n
c_1 \frac {\log n}{\sqrt n} + c_2 \frac {\log \log n}{\sqrt n} + \frac {Y_n}{\sqrt n}
, where the random variable
Y
n
Y_n
is tight, and it is attained by a triangulation that consists of
1
4
log
n
+
O
P
(
log
n
)
\frac 14\log n + O_{P}(\sqrt {\log n})
vertices. Moreover, for disk triangulations that are canonical, in that no inner triangle contains all but
O
(
1
)
O(1)
of the vertices, the minimum weight has the above form with the law of
Y
n
Y_n
converging weakly to a shifted Gumbel. In addition, we prove that, with high probability, the minimum weights of a homological filling and a homotopical filling of the cycle
(
123
)
(123)
are both attained by the minimum weight disk triangulation.