Equation of a Sphere Calculator

In three-dimensional Cartesian coordinate space, a sphere is defined as the set of all points equidistant from a fixed center point (h, k, l). The standard equation of a sphere is expressed as (x - h)² + (y - k)² + (z - l)² = r², where r is the radius. When expanded, this yields the general form: x² + y² + z² + CE·x + CF·y + CG·z + CH = 0, where the coefficients relate to the center coordinates and radius through formulas such as CE = -2h, CF = -2k, CG = -2l, and CH = h² + k² + l² - r². Surface area is computed as 4πr², and volume is calculated using the standard formula (4/3)πr³.

Imagine you need to find the equation of a sphere whose diameter endpoints are given by point 1 at (2, 4, 6) and point 2 at (6, 8, 10). First, find the center coordinates by taking the midpoint of each axis: h = (2 + 6) / 2 = 4, k = (4 + 8) / 2 = 6, and l = (6 + 10) / 2 = 8. The center is located at (4, 6, 8). Next, calculate the diameter using the 3D distance formula: d = √((6-2)² + (8-4)² + (10-6)²) = √(16 + 16 + 16) = √48 ≈ 6.928. The radius is half of the diameter, so r = 3.464 (or r² = 12). Plugging these values into the standard equation yields (x - 4)² + (y - 6)² + (z - 8)² = 12.

When working with 3D coordinate geometry problems, always double-check your input signs to avoid misplaced center coordinates. Ensure your coordinate units are consistent across all axes (x, y, and z) before initiating the calculation. If you are converting from the general equation back to the standard form, remember to complete the square carefully for each variable term to isolate the correct radius.