When a cone and a plane intersect in the right way, they form a twodimensional geometric figure known as an ellipse. An ellipsoid is the threedimensional equivalent of an ellipse; it can be visualized as a flattened sphere. The volume of an ellipsoid may be calculated from the length of its three axes.
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Express the ellipsoid in an xyz Cartesian coordinate system. The standard equation of an ellipsoid is given by x^2/a^2 + y^2/b^2 + z^2/c^2 = 1. This form has the center of the ellipsoid at the center of the coordinate system and has the axes of the ellipsoid aligned with the axes of the 3D coordinate system.

Determine the radii of the ellipsoid. This is given by the variables a, b and c in the equation from Step 1. The variable a is the radius along the xaxis, b is the radius along the yaxis and c is the radius along the zaxis.

Calculate the volume of the ellipsoid. Use the equation V = 4/3 pi x abc, where a, b and c are the three radii of the ellipsoid. For example, suppose you have the ellipsoid with the equation x^2/2^2 + y^2/3^2 + z^2/5^2 = 1. The volume of this ellipsoid would be 4/3 x pi x (2)(3)(5) = 4/3 pi (30) = 40 pi.
Tips & Warnings
 In the case of a sphere, a = b = c = r, where r is the radius of the sphere. The equation V = 4/3 pi x abc reduces to V = 4/3 pi r^3, which is the volume of a sphere.