update: when using Blender 2.73 or later make sure you get the latest version (20150206 or newer) from GitHub.
In a previous article I showed a simple addon to create a basket arch with a given width. This update adds the possibility to set the height as well. A height of 0 will give you the classic basket arch and it will silently ignore width/height ratios > 4 or < 2 (in which case it will revert to a classic arch).
For width/height ratios close to 1.0 the fit isn't perfect and although I followed the instructions on ThisIsCarpentry.com (pdf) closely the curves for the classic arch and the one with known height are not exactly the same although to be honest I not 100% sure what to expect. It might also be due to numerical inaccuracies.
In the end you might be better of scaling the arch in the z-direction.
The previous article contains source code link and installation instructions.