The equation of a sphere describes every point at distance R from the centre. It is used in 3D modelling, GPS (each satellite defines a sphere of possible positions) and physics.
How to use the Equation of a Sphere Calculator
- Select equation
- h (centre x)
- k (centre y)
- l (centre z)
- R (radius)
- E
- F
- G
- H
- A – x
- A – y
- A – z
- B – x
- B – y
- B – z
- P – x
- P – y
- P – z
Results update as you type, and everything is calculated in your browser.
How it works
From the expanded form, completing the square gives the centre (−E/2, −F/2, −G/2) and R² = E²/4 + F²/4 + G²/4 − H. A diameter AB gives the centre as its midpoint.
Inputs explained
| Field | What to enter |
|---|---|
| Select equation | Choose one: Standard form (h, k, l, R), Expanded form (E, F, G, H), Endpoints of a diameter A, B, Centre and a point P on the sphere |
| h (centre x) | A number |
| k (centre y) | A number |
| l (centre z) | A number |
| R (radius) | A number |
| E | A number |
| F | A number |
| G | A number |
| H | A number |
| A – x | A number |
| A – y | A number |
| A – z | A number |
| B – x | A number |
| B – y | A number |
| B – z | A number |
| P – x | A number |
| P – y | A number |
| P – z | A number |
Example values
How the main result changes when h changes and everything else stays at the example values above:
| h (centre x) | Centre – x |
|---|---|
| 0.5 | 0.5 |
| 0.75 | 0.75 |
| 1 | 1 |
| 1.25 | 1.25 |
| 1.5 | 1.5 |
| 2 | 2 |
Formula
Standard: (x − h)² + (y − k)² + (z − l)² = R²
Expanded: x² + y² + z² + Ex + Fy + Gz + H = 0 with E = −2h, F = −2k, G = −2l, H = h² + k² + l² − R²
Worked example
Inputs — Select equation: Standard form (h, k, l, R); h (centre x): 1; k (centre y): -2; l (centre z): 3; R (radius): 4.
Results — Centre – x = 1; Centre – y = -2; Centre – z = 3; Radius = 4; Diameter = 8; Surface area 4πR² = 201.06193; Volume 4/3·πR³ = 268.082573; Expanded-form constant H = -2.
Key terms
- Sphere
- All points at a fixed distance R from a centre.
Frequently asked questions
How do I find the centre from the expanded form?
Centre = (−E/2, −F/2, −G/2).
What if E²/4 + F²/4 + G²/4 − H is negative?
The equation describes no real sphere.
How do I get the sphere from a diameter?
The centre is the midpoint and the radius half the length of the diameter.
What is the equation of a sphere at the origin?
x² + y² + z² = R².