Sphere Equation Calculator

What Is the Equation of a Sphere?

The equation of a sphere describes all points in 3D space that lie at a fixed distance $r$ (the radius) from a center $(h, k, l)$. It extends the familiar circle equation into three dimensions.

Sphere equations are used in multivariable calculus, computer graphics, physics, and engineering. See also our Sphere Calculator and Circle Equation Calculator.

Standard Form

$$(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2$$

Here $(h, k, l)$ is the center and $r$ is the radius. When the center is at the origin, the equation simplifies to $x^2 + y^2 + z^2 = r^2$.

Expanded Form

Expanding and collecting terms gives:

$$x^2 + y^2 + z^2 + Ex + Fy + Gz + H = 0$$

Completing the square yields the center and radius:

$$h = -\\frac{E}{2}, \\quad k = -\\frac{F}{2}, \\quad l = -\\frac{G}{2}, \\quad r = \\sqrt{\\frac{E^2 + F^2 + G^2}{4} - H}$$

Example

A sphere with center $(3, 7, 5)$ and radius $10$ has equation $(x - 3)^2 + (y - 7)^2 + (z - 5)^2 = 100$.