Equation of Sphere

The equation of a sphere with its center at (a, b, c) and a radius R is:

(x - a)² + (y - b)² + (z - c)² = R²

Every point (x, y, z) on the sphere's surface satisfies it. This calculator takes the center, (X1, Y1, Z1), and any point on the surface, (X2, Y2, Z2), and finds the radius, which completes the equation, along with the surface area and volume.

How it works

The radius is the distance from the center to the point on the surface:

R = √((X2 - X1)² + (Y2 - Y1)² + (Z2 - Z1)²)

From the radius:

• Surface area: A = 4 * π * R²
• Volume: V = 4/3 * π * R³

Example

A sphere with its center at (1, -2, 3) passing through (3, 2, 7):

1. Radius:

   R = √(2² + 4² + 4²) = √36 = 6

2. The equation: with the center (1, -2, 3) and R² = 36:

   (x - 1)² + (y + 2)² + (z - 3)² = 36

3. Surface area and volume:

   A = 4 * π * 36 = 452.4 and V = 4/3 * π * 216 = 904.8

The general form

Multiplying the brackets out gives the general form:

x² + y² + z² + D * x + E * y + F * z + G = 0

For the example above, x² + y² + z² - 2x + 4y - 6z - 22 = 0. To go back, the center is (-D / 2, -E / 2, -F / 2), and the radius R = √(D² / 4 + E² / 4 + F² / 4 - G).

Frequently asked questions

How do you write the equation of a sphere from its center and radius?

Put them straight into the standard form. A sphere centered at (2, 6, -4) with a radius of 9 is (x - 2)² + (y - 6)² + (z + 4)² = 81.

Where does a sphere cross a coordinate plane?

Set that coordinate to 0. For the xy-plane, z = 0 turns the equation above into (x - 1)² + (y + 2)² + 9 = 36, a circle of radius √27 around (1, -2). If the result needs a negative radius squared, the sphere doesn't reach that plane.