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Math & Statisticsgeometry

Sphere Calculator

Sphere properties from any one of five measurements. The surface area is the derivative of the volume with respect to radius, which is why growing a sphere by a thin shell adds volume equal to the area times the thickness.

Also called: sphere volume calculator, ball volume.

Volume
4,188.79

Radius 10, diameter 20, volume 4,188.79 and surface area 1,256.64. The great circle is 62.83 around. The surface area is the derivative of the volume with respect to radius, so a thin shell adds area times thickness.

Radius
10
Diameter
20
Surface area
1,256.64
Great circle circumference
62.83
Cross section area
314.16
On the relationships
The surface area is the derivative of the volume with respect to radius, so a thin shell adds area times thickness.
How this is calculated
Method and background

How this is calculated

Everything solves back to the radius and then forward. The relationship worth noticing is that differentiating the volume formula gives exactly the surface area formula: adding a shell of thickness dr adds volume 4 pi r squared dr, which is the area times the thickness. The same holds for a circle's area and circumference, and it is the reason those formulas look related.

the surface area is exactly the derivative of the volume with respect to radius, which is not a coincidence
r
Radius

Worked examples

Each of these is asserted on every build. If a change to the engine ever moved one of these answers, the build would fail before the page could print it.

a radius of 10

Measurement
10
That is the
Radius

Volume4,188.79

4/3 pi 1000; 4 pi 100

Open this example

solving back from volume

Measurement
4,188.79
That is the
Volume

Volume4,188.79

boundary: the inverse of the first case

Open this example

Method and limits

What it assumes

  • A perfect sphere.

What it deliberately does not model

  • Units are whatever you enter, consistently: a radius in metres gives a volume in cubic metres.

Formula version 1.0.0 · definition 1.0.0 · United States · Report a problem with this calculator

Frequently asked questions

Why is the surface area formula the derivative of the volume formula?
Because growing the radius by a tiny amount adds a shell whose volume is the surface area times the thickness. The same relationship holds between a circle's area and its circumference.