Equation of a Sphere Calculator

The standard equation of a sphere is (x−h)² + (y−k)² + (z−l)² = r², where (h, k, l) is the center of the sphere and r is its radius. Any point (x, y, z) satisfying this equation lies exactly on the surface of the sphere.

The equation comes from the 3D distance formula. The distance from any surface point (x, y, z) to the center (h, k, l) must equal the radius r. Squaring both sides of the distance formula gives (x−h)² + (y−k)² + (z−l)² = r².

Expanding the standard form gives x² + y² + z² − 2hx − 2ky − 2lz + (h² + k² + l² − r²) = 0. This is often written as x² + y² + z² + Dx + Ey + Fz + G = 0, where D = −2h, E = −2k, F = −2l, and G = h² + k² + l² − r².

First, find the center by computing the midpoint of the two endpoints: h = (x₁+x₂)/2, k = (y₁+y₂)/2, l = (z₁+z₂)/2. Then compute the radius as half the distance between the endpoints. Plug these values into the standard form equation.

Use the 3D distance formula to find the radius: r = √[(x−h)² + (y−k)² + (z−l)²], where (x, y, z) is the known surface point and (h, k, l) is the center. Then substitute h, k, l, and r into the standard sphere equation.

Once you know the radius r from the equation, surface area = 4πr² and volume = (4/3)πr³. This calculator computes both automatically after deriving or receiving the radius.

Complete the square for each variable. Rewrite x² + Dx as (x + D/2)² − (D/2)², and similarly for y and z. The resulting standard form reveals the center as (−D/2, −E/2, −F/2) and the radius as √((D/2)² + (E/2)² + (F/2)² − G).

No. If after completing the square the right-hand side (r²) is negative, there is no real sphere — the equation has no real solution. If it equals zero, the 'sphere' degenerates to a single point (the center).