# Arbitrary polyhedral volume

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## Arbitrary Polyhedral Volume

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.