Get the surface area of a polyhedron (3D object)

Question

I have a 3D surface, (think about the xy plane). The plane can be slanted. (think about a slope road).

Given a list of 3D coordinates that define the surface(Point3D1X, Point3D1Y, Point3D1Z, Point3D12X, Point3D2Y, Point3D2Z, Point3D3X, Point3D3Y, Point3D3Z, and so on), how to calculate the area of the surface?

Note that my question here is analogous to finding area in 2D plane. In 2D plane we have a list of points that defines a polygon, and using this list of points we can find the area of the polygon. Now assuming that all these points have z values in such a way that they are elevated in 3D to form a surface. My question is how to find the area of that 3D surface?

Answer 1

I upvoted a few answers which I think are correct. But I think the simplest way to do it-- regardless of whether it's in 2D or 3D, is to use the following formula:

area = sum(V(i+1) × V(i))/2;

Where × is the vector cross.

The code to do this is:

    public double Area(List<Point3D> PtList)
    {

        int nPts = PtList.Count;
        Point3D a;
        int j = 0;

        for (int i = 0; i < nPts; ++i)
        {
            j = (i + 1) % nPts;
            a += Point3D.Cross(PtList[i], PtList[j]);
        }
        a /= 2;
        return Point3D.Distance(a,default(Point3D));
    }

    public static Point3D Cross(Point3D v0, Point3D v1)
    {
        return new Point3D(v0.Y * v1.Z - v0.Z * v1.Y,
            v0.Z * v1.X - v0.X * v1.Z,
            v0.X * v1.Y - v0.Y * v1.X);
    }

Note that the solution doesn't depend on projection to x-plane, which I think is clunky.

What do you think?