Yes, it must be 1/8th of the surface of a sphere of circumference 4*19 = 76, so that all 3 sides have a length of 19, and all 3 angles are 90 degrees. Simple.
Just think about the curved triangular surface of a "corner" piece on a 2x2x2 puzzle ball.
EDIT. The triangle could also be on an ellipsoid or other curved surface, but that would be a lot more tricky to define and figure out. So there are actually multiple solutions to this problem, which means that the question itself is actually not solvable because it doesn't provide enough information on the shape of the surface, and the person who wrote this question is not as clever as he/she thinks.