Three pairs account for almost every disagreement about face shape types. Exactly one of them has a defensible numeric criterion behind it, and it is not the pair people argue about most.
Oblong against oval
MeasurableThis is purely a length-to-width question, and physical anthropology has measured exactly that for a century as the morphological facial index: face height over cheekbone width, times 100, with five named bands and published population distributions. It is replicable and caliper-measurable.
What it will not give you is the boundary. The anthropometric scale is five bands on one continuum and no publication maps it onto the styling pair, so inventing a cutoff where oval becomes oblong would be manufacturing a number. Use the index to describe how long a face is, not to adjudicate the label.
Round against square
No published criterionThe difference here is jaw angularity, which the facial index cannot see at all: two faces with an identical index can have a soft jaw or a sharp one. The nearest real measurement is the gonial angle, the angle at the corner of the jaw, which is a genuine anthropometric variable.
But no paper defines where a square face begins in degrees, and using the gonial angle that way would be fabrication. One warning if you are tempted to reason it out yourself: the literature reports that this angle increases with age, so the folk rule that a square jaw means a low angle is not a safe simplification.
Heart against triangle
No measurement basisThere is none, because triangle is not in the research at all. The published machine-vision work on face shape uses five classes, heart, oblong, oval, round and square, and triangle appears in none of them.
Triangle and diamond are both styling-site additions, which is visible in their own numbers: three tool sites report triangle at 0.03 percent, 0.05 percent and 8.6 percent. A category whose reported prevalence varies by a factor of nearly 300 between sources is not a category anyone is measuring.