How are the physiological properties of mitochondria measured?
A simple method would me to express a fluorescent protein in the cell that specifically localizes to mitochondria. mito-dsRed is a Red Fluorescent Protein which has that property; it can be expressed from a plasmid.

Using fluorescent microscopy and image analysis, you would be able to measure the geometric properties of the mitochondria. This can be done at a single cell level too (using confocal microscope) and non-destructively. Volume can be measured by what is known as Z-stacking— an automated technique by which the microscope takes several 2D snaps at different planes and stacks them to construct a 3D image. 

You can implement this in live cells too; this method is called live cell imaging.