# Calculate surface area and outward normal of each face(non-planar) for a irregular hexahedron.

I am using the vector cross product to calculate each surface area and thus the outward normal. However, this approach fails when the surfaces are non-planar. So what is the best approach to calculate surface area and normal for a non-planar face of a hexahedron?

• BTW, if the edges are straight segments, then it's likely your hexahedron is in fact an irregular dodecahedron. Just turn the quadrangles into pairs of triangles.
– SF.
Dec 5, 2018 at 16:58

## 1 Answer

If the surface is nonplanar then the surface normal is a (hopefully continuous and differentiable) function of location. You will have to integrate the surface function S(x,y,z) over any two of the dimensions. If this is an engineering problem, then of course use a finite-grid approach and numerical integration. Then treat each grid element as a planar surface.

• Thanks, but all I have got is the coordinates of the 8 vertices. Also, this process is basically repeated over a number of hexahedrons, arbitrarily oriented, so I think the surface function option won't be feasible. This is basically a hex mesh in a finite element analysis. I was hoping to not complicate further calculations with using grids within each element to calculate surface area/normals. Dec 5, 2018 at 21:31