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Arbitrary polyhedral volume

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The volume of arbitrary polyhedral can be calculated by using Green-Gauss Theorem.

\int\limits_\Omega  {div(\vec F)d\Omega  = } \oint\limits_S {\vec F \bullet d\vec S}

By choosing the function

\vec F = \frac{{\left( {x\hat i + y\hat j + z\hat k} \right)}}{3}

Where (x,y,z) are centroid of the surface enclosing the volume under consideration. As we have,

div(\vec F) = 1

Hence the volume can be calculated as:

volume = \oint\limits_S {\vec F \bullet \hat ndS}

where the normal of the surface pointing outwards is given by:

\hat n = (n_x \hat i + n_y \hat j + n_z \hat k)

Final expression could be written as

volume = \frac{1}{3}\sum\limits_{faces} {\left[ {\left( {x \times n_x  + y \times n_y  + z \times n_z } \right) \bullet S} \right]}

where S is magnitude of Surface Area.

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