Hang it up, throw darts and when someone shouts "Bullseye"........
I'll go away now......
OK, random thoughts/guesses:-
Simple plastic ruler measurements from the edge? Being concave, the ruler will sit above the surface when rested on the edges.
Put it on a flat and horizontal (check) surface, add a small drop of water with a little detergent to break the surface tension to the approximate centre. A few vigorous taps and the drop should settle at the lowest point?