Is this a design problem? As in, you are supposed to choose a concave shape, and then calculate the length?

A parabola, a hyperbola, a circular arc, and an elliptical arc are all conics that can be concave and can fit there. Once you have the shape (whether it be given, or whether you design it yourself), just write the equation as y = f(x). x is the vertical distance, and y is the horizontal distance. Then the ? is just f(h).

For example, the surface of the top shape is described by the equation y = ax. Substitute x = h, and y(h) = ah.