If you have a region enclosed by a boundary contour, sure one easy way to compute the area inside of it is to do a “scan-fill” of the region and count the total number of pixels or voxels, but what about a resolution-independent vector computation approach? The obvious first step would be to tesselate down the region into triangles or tetrahedrons, for which the area can be computed trivially in a vector manner. But how do you get there? One approach that I thought of is by progressive convex hull determination. First you determine the convex hole of the entire region, calculate the area of that, then you subtract convex hulls that cut into it to make a concave region. If you have additional cuts on each of those, you add back the area of those convex hulls. Keep doing this until you recurse down to the last convex sub-shapes.
Another, purportedly more popular, alternative is to compute the Delaunay triangulation of the region, then simply add up the area or volume of those triangles. Except for the triangulation part, it is otherwise easy and simple, conceptually speaking.
20190114/https://en.wikipedia.org/wiki/Delaunay_triangulation
20190114/https://en.wikipedia.org/wiki/Constrained_Delaunay_triangulation
20190114/https://en.wikipedia.org/wiki/Chew%27s_second_algorithm
20190114/https://en.wikipedia.org/wiki/File:LakeMichiganMesh.png