Equation Of A Sphere Calculator

Work out equation of a sphere instantly with clear inputs, formula shown and shareable results.

About Equation Of A Sphere Calculator

A sphere is the set of points at a fixed distance r from a centre (a, b, c), which the distance formula turns into (x − a)² + (y − b)² + (z − c)² = r².

Formulas

Standard form

(x − a)² + (y − b)² + (z − c)² = r²

Expanded form

x² + y² + z² − 2ax − 2by − 2cz + (a² + b² + c² − r²) = 0

Worked example

With the default values below, Equation Of A Sphere Calculator returns Equation of the sphere of (x - 1)^2 + (y + 2)^2 + (z - 3)^2 = 25. Change any input above and every figure here is recalculated in your browser.

Inputs used

Centre (x, y, z): 1, -2, 3
Radiusminimum 0: 5

Results

Equation of the sphere: (x - 1)^2 + (y + 2)^2 + (z - 3)^2 = 25
r²: 25.0000
Volume: 523.5988
Surface area: 314.1593

Step by step

1Values used
2Standard form
3Expanded form
4Equation of the sphere= (x - 1)^2 + (y + 2)^2 + (z - 3)^2 = 25
5r²= 25.0000
6Volume= 523.5988
7Surface area= 314.1593

Assumptions and limits

The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 4 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

How do I read the centre off an expanded equation?

Halve and negate each linear coefficient: the x term −2a gives a, and so on. Then complete the square to find r².

What if r² comes out negative?

There is no real sphere; the equation describes an imaginary sphere or a single point when r² = 0.