Equation of a Sphere Calculator

Enter a sphere's center coordinates and radius to get its equation in standard form (x-a)²+(y-b)²+(z-c)²=r², its expanded general form, surface area, and volume.

Quick Facts

Standard form
(x−a)² + (y−b)² + (z−c)² = r²
Center (a, b, c), radius r.
General form
x²+y²+z²+Dx+Ey+Fz+G=0
D=−2a, E=−2b, F=−2c, G=a²+b²+c²−r².
Surface area
A = 4πr²
Depends only on the radius.
Volume
V = (4/3)πr³
Depends only on the radius.

Your Results

Calculated
Standard Equation
-
(x−a)² + (y−b)² + (z−c)² = r²
General Form Equation
-
x²+y²+z²+Dx+Ey+Fz+G=0
Surface Area
-
A = 4πr²
Volume
-
V = (4/3)πr³

Ready

Enter the center coordinates and radius, then press Calculate.

Formula and Method for the Equation of a Sphere

A sphere is the set of all points in three-dimensional space that lie exactly a fixed distance — the radius r — from a fixed point, the center (a, b, c). Applying the distance formula in 3D directly gives the standard equation of a sphere: (x − a)² + (y − b)² + (z − c)² = r². This calculator takes a center and radius and derives that standard equation, its expanded general form, the sphere's surface area, and its volume.

How the calculation works

Starting from (x − a)² + (y − b)² + (z − c)² = r², expanding each squared binomial and collecting terms gives the general form: x² + y² + z² + Dx + Ey + Fz + G = 0, where D = −2a, E = −2b, F = −2c, and G = a² + b² + c² − r². This is the same sphere written a different way — useful because many textbook and software problems present a sphere's equation already expanded and ask you to find its center and radius. To reverse the process, complete the square on each variable: center = (−D/2, −E/2, −F/2) and radius = √(D²/4 + E²/4 + F²/4 − G). Surface area and volume depend only on the radius, not the center's position: A = 4πr² and V = (4/3)πr³.

Common mistakes

  • Sign errors on the center: the standard form subtracts the center coordinates, so a center of (−3, 2, 0) produces (x + 3)² + (y − 2)² + z² = r², not (x − 3)².
  • Radius vs. radius squared: the right-hand side of the standard equation is r², not r. A sphere of radius 5 has "= 25" on the right side, not "= 5".
  • Units on area and volume: surface area is in square units (ft², m²) and volume is in cubic units (ft³, m³) — never report either with the plain linear unit.

Real-world applications

  • 3D modeling, CAD, and computer graphics use the sphere equation for bounding volumes, collision shapes, and rendering primitives.
  • Physics uses spherical equations to describe fields and potentials around point sources (gravity, electric charge, radiation).
  • Robotics and game development use sphere equations for fast collision and proximity detection.
  • GPS trilateration and satellite coverage modeling rely on spheres centered on known points with known radii.

Frequently Asked Questions

What is the standard equation of a sphere?
A sphere with center (a, b, c) and radius r is described by (x − a)² + (y − b)² + (z − c)² = r². Any point (x, y, z) that satisfies this equation lies exactly r units from the center, which is the definition of a sphere.
How do I convert the general form back to center and radius?
Given x² + y² + z² + Dx + Ey + Fz + G = 0, complete the square on each variable. The center is (−D/2, −E/2, −F/2) and the radius is r = √(D²/4 + E²/4 + F²/4 − G). If that quantity under the square root is negative, the equation does not describe a real sphere.
How do I find the surface area and volume of a sphere?
Surface area is A = 4πr² and volume is V = (4/3)πr³, using only the radius. For a sphere of radius 5, surface area is 4π(5²) ≈ 314.16 square units and volume is (4/3)π(5³) ≈ 523.60 cubic units.
Why does the calculator require a positive radius?
A radius of zero collapses the equation to a single point, not a sphere, and a negative radius has no geometric meaning since distance cannot be negative. Enter r > 0 to get a valid sphere equation.

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