I suppose a silly answer is to immerse it in liquid and measure the volume of the displaced liquid???????
An octahedron can be distorted in many ways. If at least one twofold axis or a mirror plane or an inversion center remains after the deformation, then you can divide the volume into two tetragonal pyramids with flat basis. The volume of the pyramids can then easily be calculated i.e 1/3 of the product of the basis multiplied by the height. If the pyramid has no symmetry, then the method of the tetrahedra is possible.
Thanks Chapuis and Horvath,
Chapuis,
Do you mean I have to apply below mentioned formula for calculating the volume of Tetrahedron for each pyramid and sum it up ??
V = (a^3) / 6 x square root of 2
For a regular tetrahedron with all edges equal, the volume V is given by a^3*sqrt(2)/12 (beware, not by 6). In case of a deformed tetrahedron with all edges different, this formula is not valid. The correct formula is given in
http://mathcentral.uregina.ca/QQ/database/QQ.09.06/haivan1.html
An irregular octahedron can be dissected into four irregular tetrahedra. Volume of any tetrahedron is a third the base area times the perpendicular height. Doesn't have to be regular. Its base is a triangle, area is half the base edge length times the perpendicular height. Doesn't have to be regular.
1. There exists a formula for calculate the volume of an arbitrary octahedron in terms of edges lengths. This formula was proved by Astrelin and Sabitov, see [Russian Mathematical Surveys, 1995, 50:5, 1085–1087].
2. If you know coordinates of vertices of an octahedron, then you can easy calculate the volume using the cross product of vectors going from an arbitrary point to the vertices.
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