Leaving introductory geometry behind, here is an illuminating explanation for how to draw on differential geometry and other helpful tools to calculate the volume of a mesh, from Jesse of nervous system — one of Adafruit’s favorite design teams as well as an excellent resource to learn more about leveraging good math to produce good art:

In order to generate the price of a custom design on the fly, we need to calculate the volume of the piece for 3d printing. By constantly updating the volume, the customer gets instant feedback on how their changes are affecting price. Calculating the volume of a mesh is a relatively simple and well-known problem, and I’ll go over the straight forward case as well as an optimization we’ve incorporated into our latest project.

…Now, onto the good stuff. What happens if you have an object that is made of (at least in part) an aggregate of a bunch of identical but complex parts. I don’t mean a booleaning together primitives, but you could imagine something like a buckyball where each face is articulated with some kind of intricate shape. The brute force approach would be to move and rotate the shape to the proper position then go through each triangle and calculate the volume. This means you have to calculate a transform on each of the points of your shape, and then go through each triangle. If your shape has 1000 triangles and you have 100 shapes, that ends up being a lot calculation. We can drastically increase the efficiency of this by computing a “general volume” for the shape once, and applying our transforms only to that simplified representation. But what does this general volume look like?

The key idea behind this general volume is the fact that volume is rotation invariant. This is one of the basic results of differential geometry. It is intuitively obvious; no matter how I orient an object in space its volume does not change. What is less intuitive is that the same thing holds true for the signed volume of open shapes. Mathematically this can be seen easily by noting that the volume is the determinant of a matrix, and the rotation matrix has a determinant of 1. The determinant of one matrix multiplied by another is the multiplication of their individual determinants. So, I can rotate my primitive element however I want, and the volume stays the same. If I was only rotating my shape, then I could calculate the volume of my shape once and multiply it by the number of shapes I have.