The net for a triangular prism consists of 2 identical triangles, here all nets have sides 5,12, and 13. 3 rectangles, all with a common dimension: the height H=14 of the prism.
Each of the 3 rectangles should have dimensions H=14 plus one of the following: each of 5, 12, 13 corresponding to one side of the base (triangle).
So the dimensions of the rectangles are 5x14, 12x14, 13x14
And dimensions of the triangles are 5,12 and 13.
The total surface area is therefore (5+12+13)*14 + 2*(5*12/2) =420+60 =480.
There is only one net that satisfies this condition.
