There is an interesting discussion here: vector - Calculating roundness/compactness of polygon? - Geographic Information Systems Stack Exchan... and some more background here: Roundness (object) - Wikipedia .
Is that what you are looking for?
'circularity' various definitions: aka roundness
You can calculate some of the volume properties using
Bounding Containers ArcMap has an equivalent toolset (ie convex hull, MABE, aligned rectangle, minimum area circle)
Concave Hull The ratio of the convex hull to the concave hull is often a measure of container tortuosity.