How can we find missing lengths using geometric properties of length, proportion, and geometric mean?

I would need to see a sample problem in order to be much more specific. Generally speaking, the Pythagorean Theorem is probably the most used method for finding lengths. If you know the width and height of an object, the distance between opposite corners is the square root of the sum of the square of the height plus the square of the width.

For more complicated shapes, good estimates can be found my taking the geometric mean between the lengths of an inner circumscribed path and an outer circumscribed path. Often the path is refined in a repeated way until any remaining error in the estimated length comes within a specified tolerance.

I hope this helps!