How to Find the Equations of Spheres

You can describe spheres using equations.

If you know a sphere's radius and the coordinates of its center, you can write the equation of the sphere in three dimensions using the variables X, Y and Z. Knowing how to express a sphere in mathematical notation is useful if you study multi-variable calculus or need to solve physics problems involving spheres and coordinate axes. Also, in order to graph a sphere with mathematical modeling software, you must input the sphere as an equation.

Instructions

1. Determine the radius of the sphere and call this number R. For instance, if a sphere's diameter is 10, then R = 5.

2. Determine the x-coordinate of the sphere's center and call this number A. If you are given the center of a sphere as a coordinate point with three components, then the first number is A. For instance, if the sphere is centered at (3,4,9), then A = 3.

3. Find the y-coordinate of the center and call this number B. For example, if the center is at the point (3,4,9), B is the middle number in the set, so B = 4.

4. Find the z-coordinate of the sphere's center and call this number C. As in the example, if the center is (3,4,9), C is the last component, so C = 9.

5. Put all the variables X, Y and Z and numbers R, A, B and C in the sphere equation, R^2 = (X-A)^2 + (Y-B)^2 + (Z-C)^2. Because R = 5, A = 3, B = 4 and C = 9 from the example, the equation of the sphere is 25 = (X-3)^2 + (Y-4)^2 + (Z-9)^2. This is a three-dimensional equation in three variables.

References

• University of Southern Mississippi; Three-Dimensional Coordinate Systems; Jim Lambers