Surface Area of Arbitrary 3D Geometry

Download: area.zip

• Surface Area of Triangle
• Surface Area of 3D Geometry
• Approximation Error

Surface Area of a Triangle

Suppose there is a single triangle V₁-V₂-V₃ in a 3D space. The surface area of this triangle is a half of the parallelogram constructed by V₁, V₂ and V₃.

where a is the length of V₁-V₂, and b is the length of V₁-V₃.

So, you can get the area by performing 2 square roots and sine function. But, there is a simpler formula using the cross product of 2 vectors.

By definition of cross product, the magnitude of the cross product is equal to the area of parallelogram constructing 2 vectors.

The cross product is decomposed to a sum of 3 bases vectors and it's magnitude is

Therefore, the area of the triangle is a half of the parallelogram's area, which is the magnitude of the cross product.

Also note that the cross product can be expressed as a matrix notation, and the cross product is equivalent to the determinant of the matrix.

Surface Area of 3D Geometry

If an arbitrary 3D geomery is triangulated, the surface areas of the geometry is the sum of area of all triangles constructing the geomery.

If a geomery consists of N triangles, the total surface area is;

Here is C++ code snippet to compute the surface area of 3D geometry. The input parameters are an array of vertices, an array of indices and the number of indices for the geometry. See more detail in geomUtils.cpp file.

// calculate area of a single triangle
double computeTriangleArea(float v1[3], float v2[3], float v3[3])
{
    float v12[3];           // v2 - v1
    v12[0] = v2[0] - v1[0];
    v12[1] = v2[1] - v1[1];
    v12[2] = v2[2] - v1[2];
    float v13[3];           // v3 - v1
    v13[0] = v3[0] - v1[0];
    v13[1] = v3[1] - v1[1];
    v13[2] = v3[2] - v1[2];

    // v12 * v13 (cross product)
    double cross[3];
    cross[0] = v12[1]*v13[2] - v13[1]*v12[2];
    cross[1] = v12[2]*v13[0] - v13[2]*v12[0];
    cross[2] = v12[0]*v13[1] - v13[0]*v12[1];

    return 0.5 * sqrt(cross[0]*cross[0] + cross[1]*cross[1] + cross[2]*cross[2]);
}

// calculate area of a geometry
double computeArea(float* vertices, unsigned int* indices, int indexCount)
{
    double sum = 0;
    for(int i = 0; i < indexCount; i += 3)
    {
        float* v1 = &vertices[indices[i]  *3];
        float* v2 = &vertices[indices[i+1]*3];
        float* v3 = &vertices[indices[i+2]*3];
        sum += computeTriangleArea(v1, v2, v3);
    }
    return sum;
}

Approximation Error

Since a 3D geometry is represented with a limited number of triangles, there is an approxiamtion error. For example, the exact area of a sphere with its radius 1 is 12.56637061..., but the area of the sphere with only 1,000 triangles is 12.4654 (99.2% of the exact area). The following chart shows the approximation errors while the number of triangles increases. A sphere with 1M triangles is close to the exact area upto 4 decimal places.

Number of Triangles	Computed Area	Error	Accuracy
1,000 tris	12.4654	0.10095	99.1966%
5,000 tris	12.5439	0.02248	99.8211%
10,000 tris	12.5563	0.01009	99.9197%
50,000 tris	12.5642	0.00218	99.9826%
100,000 tris	12.5653	0.00104	99.9917%
500,000 tris	12.5661	0.00023	99.9982%
1,000,000 tris	12.5663	0.00010	99.9992%

Download the testing program to calculate the surface area and error for a sphere. You can increase the number of triangles by pressing SPACE key.
Download: area.zip