I have set of three dimensional coordinates, and the shape is unknown. I would like to calculate the surface area and volume for these coordinates approximately. What is the right approach to solve this?
2026-03-25 23:39:34.1774481974
How to calculate volume and surface area of three dimensional figures given set of three dimensional coordinates?
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Although I have done similar problems in two dimensions, I have never done this in three.
But it seems to me that the problem of finding the surface area and volume of the convex hull (convex polyhedron) of a given finite set of points defined by their coordinates is do-able but difficult. The hard step is the first: given the points, find the triangles that make up the sides of the polyhedron. (A side may be a polygon with more than three sides, but that polygon can be broken up into triangles, so the triangles can be found for any convex polyhedron.) Given that, finding the surface area and volume are not difficult.
However, I have no idea for a practical algorithm to find the bounding triangles for the polyhedron in the general case. An impractical algorithm for many points is to check every combination of three non-collinear points from the set, and see if all the other points lie on one side of the plane defined by those points. If so, that would be a bounding triangle. You would need to check for co-planar triangles resulting from this, and find non-overlapping triangles for the polygon. There are many sites that explain how to do polygon triangulation.