# How do I calculate surface area given a three dimensional coordinates of a face?

I have three dimensional coordinates of a face, how do I calculate surface area?

• What kind of face do you mean? – Matthias Sep 22 '14 at 15:07
• It can be any shape, but in the simpler case I can assume it to be a rectangle face. – user10550 Sep 22 '14 at 15:09
• Likely you have faces of a polyhedron in mind (it's worth stating this explicitly). One difficulty is that if the coordinates are not represented exactly, a face with more than three vertices may fail to lie exactly in a plane, due to rounding errors. – hardmath Sep 22 '14 at 15:26

In general for a parametized face $\vec F(u,v)$ the area is the integral $$A=\int \left|\frac{\text d\vec F(u,v)}{\text du}\times\frac{\text d\vec F(u,v)}{\text dv}\right|\text du \text dv$$