# Equation of a Sphere Calculator

URL: https://speedycalculator.com/calculators/math/sphere-equation/
Last updated: 2026-10-30

> The equation of a sphere calculator converts between standard form (x − h)² + (y − k)² + (z − l)² = R² and expanded form, or builds the sphere from a diameter or a centre and a point, and gives its radius, surface area and volume.

## Inputs

- 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

- Centre – x
- Centre – y
- Centre – z
- Radius
- Diameter
- Surface area 4πR²
- Volume 4/3·πR³
- Expanded-form constant H

## 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
```

## Explanation

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

FieldWhat to enterSelect equationChoose 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 sphereh (centre x)A numberk (centre y)A numberl (centre z)A numberR (radius)A numberEA numberFA numberGA numberHA numberA – xA numberA – yA numberA – zA numberB – xA numberB – yA numberB – zA numberP – xA numberP – yA numberP – zA number

### Example values

How the main result changes when **h** changes and everything else stays at the example values above:

h (centre x)Centre – x0.50.50.750.75111.251.251.51.522

## 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.

## FAQ

**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.


## Sources

- [OpenStax – Calculus Volume 3](https://openstax.org/details/books/calculus-volume-3)
- [Wolfram MathWorld – Sphere](https://mathworld.wolfram.com/Sphere.html)