Sphere equation

This calculator accepts those four numbers — the three center coordinates and a strictly positive radius — and returns the textbook identity (x − h)² + (y − k)² + (z − l)² = r² together with the expanded general form x² + y² + z² + Dx + Ey + Fz + G = 0. The four coefficients D = −2h, E = −2k, F = −2l, G = h² + k² + l² − r² and the radius squared travel in the same payload so a student, a grader, or a CAD script can copy either form without expanding by hand. Radius must be a positive finite number; a missing, zero, negative, or non-numeric value fails with invalid_input instead of a silent NaN. Through the API a successful request costs $0.002; validation failures are not billed.

What the standard equation of a sphere is and which coefficients this tool returns

A sphere in three-dimensional Cartesian space is completely determined by a center point (h, k, l) and a positive radius r: it is the set of all points whose Euclidean distance from that center is exactly r. Squaring both sides of the distance formula removes the square root and produces the standard equation (x − h)² + (y − k)² + (z − l)² = r², which is the form textbooks ask students to write down and the form most computer-graphics and CAD notes start from. A translated sphere such as center (1, 2, 3) and radius 5 becomes (x − 1)² + (y − 2)² + (z − 3)² = 25, with coefficients D = −2, E = −4, F = −6 and G = −11. Expanding the three squared binomials and collecting like terms yields the general form x² + y² + z² + Dx + Ey + Fz + G = 0, whose four coefficients are D = −2h, E = −2k, F = −2l and G = h² + k² + l² − r². This calculator returns both writings of the same surface, the four coefficients, and the radius squared, so you can paste the standard identity into a homework line or feed D, E, F, G into a solver that prefers the monic general form. The unit sphere at the origin is the special case h = k = l = 0 and r = 1, which collapses to x² + y² + z² = 1 and to x² + y² + z² − 1 = 0. Radius must be strictly positive: r = 0 is a single point, not a sphere, and a negative radius is not a geometric length. Those cases fail with invalid_input rather than a hollow success payload. This tool does not compute volume or surface area, does not fit a sphere through four sample points, and does not convert units; the search intent is the algebraic equation of a sphere from a known center and radius, with a clear answer or an honest error.

Input contract, coefficient identities, and how the two equation strings are built

After parsing, the engine insists that r is strictly greater than zero and that every coordinate stays inside the published absolute bound of 10^12, so a non-positive radius never becomes a fake r² and an overflow never becomes a silent Infinity. Once the four values pass, radius squared is r times r, D is −2h, E is −2k, F is −2l and G is the sum of the three squared center coordinates minus that radius squared. The standard-equation string writes each axis with a sign-aware binomial: a zero center prints x² rather than (x − 0)², a positive center prints (x − h)², and a negative center prints (x + |h|)², and likewise for y and z. The general-equation string always starts from x² + y² + z² and then appends only the non-zero linear and constant terms, so the unit sphere at the origin is x² + y² + z² − 1 = 0 rather than a cluttered line of plus-zero coefficients. POST a JSON body with four required finite numbers: h, k and l for the center, and r for the radius. Each field may arrive as a JSON number or as a numeric string; aliases cx or center_x, cy or center_y, cz or center_z, and radius are accepted so a CAD-style payload does not need remapping. Missing fields, empty strings, NaN and Infinity fail with invalid_input naming the field. Pair the free browser run on this page with the priced API when you need the identical algebraic result inside an automated pipeline without re-expanding the three binomials yourself.

Where a sphere-equation calculator earns its keep in study, design, and code

On this page the work runs entirely in your browser; through the API each successful lookup costs $0.002, failed validation is not charged, and nothing is stored after the response returns. That split — free interactive exploration and a priced, identical API — lets you move from a classroom demo to an automated pipeline without changing the algebraic result. Whether you arrived searching for a sphere equation calculator or for the standard form from center and radius, the contract is the same: finite center, positive radius in, coefficients and r² out, or a clear invalid_input.