How to Calculate Elliptical Volume

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When a cone and a plane intersect in the right way, they form a two-dimensional geometric figure known as an ellipse. An ellipsoid is the three-dimensional 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.

Things You'll Need

  • Calculator
  • Express the ellipsoid in an x-y-z 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 3-D 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 x-axis, b is the radius along the y-axis and c is the radius along the z-axis.

  • 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.

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