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
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.
How to Convert Ellipsoid to Orthometric Height
The Earth may not be flat, but its exact shape is still a cause of debate. Since the first accurate measurements of...
How to Calculate Gallons in a Pool
Determining the amount of water in the pool is essential when adding pool chemicals and sizing pumps and filters. Chemicals are added...
How to Calculate the Volume of an Oval
Finding the volume of an oval, such as a casserole dish, is easy. Fill it up with water, pour the water into...
How to Calculate the Volume of an Ellipsoid
You can measure volume even if it's in the shape of a tear drop. A tear drop is an ellipsoid, similar to...
How to Calculate the Volume Under the Elliptic Paraboloid
An elliptic paraboloid is a three-dimensional surface that is used in calculus. It has a distinctive nose-cone appearance. The vertical cross sections...
How to Find the Volume of an Irregular Solid
Volume depends on the dimensions of an object, but can be difficult to calculate if your object is irregular. In this case,...
How to Calculate the Volume for Irregular Shapes
Calculating the volume of shapes is usually quite simple, especially when you use simple formulas for regular shapes like triangles, rectangles or...
Formula for the Volume of a Octagon
In geometry, an octagon is a polygon with eight sides. A regular octagon has eight equal sides and equal angles. The regular...