Sphere Equation Calculator — Standard & General Form
Enter the center (h, k, l) and radius r of a sphere to instantly get both the standard form equation (x−h)² + (y−k)² + (z−l)² = r² and the expanded general form, along with diameter, volume, surface area and great-circle circumference.
units
r = 5
- 1
Radius squared
5² = 25 - 2
Radius cubed
25 × 5 = 125 - 3
Volume
4⁄3 × π × 125 = 523.5988Volume of the sphere — the primary geometric quantity derived from the radius.
How does this calculator work?
A sphere centered at (h, k, l) with radius r has the standard equation (x − h)² + (y − k)² + (z − l)² = r². Expanding gives the general form x² + y² + z² − 2hx − 2ky − 2lz + (h² + k² + l² − r²) = 0. Enter center and radius to get both forms plus volume and surface area.
Formula
How this is calculated
A sphere in three-dimensional Euclidean space is the set of all points (x, y, z) whose distance from a fixed center point (h, k, l) equals the radius r. Applying the distance formula directly gives the standard equation: (x − h)² + (y − k)² + (z − l)² = r². This form makes the center and radius immediately readable.
Expanding all the squared terms and collecting constants produces the general form x² + y² + z² + Dx + Ey + Fz + G = 0, where D = −2h, E = −2k, F = −2l, and G = h² + k² + l² − r². Going the other way — recovering center and radius from a general-form equation — requires completing the square in each variable: group the x, y, and z terms, add (D/2)², (E/2)², and (F/2)² to both sides, and read off the center and r² = (D/2)² + (E/2)² + (F/2)² − G.
The calculator also computes volume (4⁄3 π r³), surface area (4 π r²) and great-circle circumference (2 π r) so you have the complete geometric picture in one place. The coordinate system is abstract — whatever unit you assign to r applies throughout.
Frequently asked questions
Group x, y, and z terms separately and complete the square in each variable. For x: x² + Dx = (x + D/2)² − (D/2)². Do the same for y and z, then move the constant terms to the right side to get r² = (D/2)² + (E/2)² + (F/2)² − G and the center (−D/2, −E/2, −F/2).
When (h, k, l) = (0, 0, 0), the standard equation simplifies to x² + y² + z² = r². The general form then has D = E = F = 0 and G = −r². Every point on the sphere is equidistant from the origin.
A circle in 2D has equation (x − h)² + (y − k)² = r² with center (h, k). Adding the third dimension (z − l)² extends this to a sphere. A circle is the intersection of a sphere with any plane through the center (a great circle) or offset (a small circle).
TG we-Calculate Editorial Team. (2026). Sphere Equation Calculator — Standard & General Form [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/sphere-equation-calculator
TG we-Calculate Editorial Team. "Sphere Equation Calculator — Standard & General Form." TG we-Calculate. 2026. https://we-calculate.com/calculator/sphere-equation-calculator.
TG we-Calculate Editorial Team, "Sphere Equation Calculator — Standard & General Form," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/sphere-equation-calculator
@misc{wecalculate_sphere_equation_calculator, title = {Sphere Equation Calculator — Standard & General Form}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/sphere-equation-calculator}}, year = {2026}, note = {TG we-Calculate} }
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