POSSIBLE SOLUTION(S)
Hausdorff distance (see Extensions)
It is difficult to obtain the parabola that is closest in the Hausdorff distance to the true arch, but it provides a nice way to judge the other solutions. Here is a summary which is shown in detail in the Extensions section, a subsection of Teacher Notes.
From best to worst using Hausdorff distance:
(1) Setting the arc lengths equal and matching the curves at the ground
(2) Minimizing the distance in the least squares sense
(3) Matching curves at the vertex and ground
(4) Setting the arc lengths equal and matching at the vertex
The four cases are shown below from best to worst fit in the Hausdorff sense.