Sphere Equation Calculator

Generate the standard and expanded equation of a sphere from center and radius.

Enter Center & Radius

Center (h, k, l)

Result

Standard Form
(x-2)² + (y+3)² + (z-1)² = 25
standard equation
Center(2, -3, 1)
Radius5
r²25
Expanded Formx² + y² + z² - 4x + 6y - 2z - 11 = 0
Volume523.598776
Surface Area314.159265

Step-by-Step Solution

(x-h)² + (y-k)² + (z-l)² = r²

Equation of a Sphere

The equation of a sphere in three-dimensional space defines all points (x, y, z) that are at a fixed distance (the radius r) from a center point (h, k, l). This is the 3D analogue of the equation of a circle.

Forms of the Sphere Equation

Standard Form

Shows the center and radius directly. Easy to read and graph.

(x-h)² + (y-k)² + (z-l)² = r²

Expanded (General) Form

Fully expanded polynomial. Useful for algebraic manipulation.

x² + y² + z² + Dx + Ey + Fz + G = 0

Unit Sphere

A sphere centered at the origin with radius 1.

x² + y² + z² = 1

How to Convert Between Forms

  1. Standard to Expanded: Expand each squared term and combine like terms.
  2. Expanded to Standard: Complete the square for x, y, and z separately.
  3. From expanded form x² + y² + z² + Dx + Ey + Fz + G = 0: center = (-D/2, -E/2, -F/2), r = sqrt((D/2)² + (E/2)² + (F/2)² - G).

Applications

Sphere equations are used in 3D graphics programming, physics simulations, GPS satellite positioning, molecular modeling, and astronomical calculations involving celestial bodies.