What this calculator does
A sphere is the simplest of the three-dimensional shapes to describe: one measurement, the radius, determines everything about it. Volume grows with the cube of the radius while surface area grows with its square, and the gap between those two growth rates is the single fact that matters most in practice.
That gap is why small spheres cool and dry faster than large ones for their size, why a large scoop of ice cream melts slower per gram than a small one, and why doubling a tank’s radius takes eight times the material to fill but only four times the material to build.
The formula
Volume comes from integrating the area of circular cross-sections through the sphere, giving 4/3πr³. Surface area, 4πr², is exactly four times the area of the sphere’s largest circular cross-section: a tidy relationship discovered by Archimedes, who considered it his best result and had it carved on his tomb.
| Term | Meaning |
|---|---|
| r | Radius: centre to surface, the only input. |
| V | Volume, the space enclosed, equal to 4/3πr³. |
| A | Surface area, equal to 4πr². |
| Great circle | The largest possible circular cross-section, passing through the centre. |
The inputs explained
| Field | What to enter |
|---|---|
| Radius | The radius of the sphere, centre to surface. |
When to use it
Sizing a ball, dome or spherical tank
Volume gives the capacity: for a tank, the litres figure converts it directly if the radius was measured in centimetres. Surface area gives the material needed to coat or clad it.
Comparing planets, balls or fruit by size
Because volume scales with the cube of the radius, a sphere twice the diameter of another has eight times the volume, not twice: the same relationship behind why Earth’s volume dwarfs the Moon’s far more than their diameters suggest.
Estimating surface area for paint, coating or heat loss
Anything proportional to surface area, paint coverage, heat radiated, drag through a fluid, follows the r² relationship, growing more slowly than volume as an object scales up.
Working from a diameter measurement instead
If you measured across the sphere rather than from its centre, halve that figure first: this calculator, like the standard formulas, needs the radius.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How volume and surface area scale with radius
A single dimension, doubled and doubled again, showing how volume pulls away from surface area.
| Radius | Volume | Surface area | Volume in litres if measured in cm |
|---|---|---|---|
| 1 | 4.189 | 12.566 | 0.00 L |
| 5 | 523.599 | 314.159 | 0.52 L |
| 10 | 4,188.79 | 1,256.64 | 4.19 L |
| 20 | 33,510.32 | 5,026.55 | 33.51 L |
| 50 | 523,598.78 | 31,415.93 | 523.60 L |
Questions
How do I find the volume of a sphere from its diameter?
Halve the diameter to get the radius first, then apply 4/3πr³. Using the diameter directly in that formula in place of the radius will overstate the volume by a factor of eight.
Why does volume grow faster than surface area as a sphere gets bigger?
Volume depends on the radius cubed while surface area depends on the radius squared. Cubing always outpaces squaring for values above 1, so the gap widens the larger the sphere becomes.
What is a great circle?
The largest circle that can be drawn on a sphere’s surface, formed by any plane passing through the centre: the equator on a globe is one example. Its circumference and area both equal those of a circle with the same radius as the sphere.
How is surface area related to the great circle?
A sphere’s total surface area is exactly four times the area of its great circle: one of the results Archimedes considered his finest and asked to have marked on his tomb.
How do I convert a sphere’s volume to a real capacity, like litres?
If the radius is measured in centimetres, the volume in cubic centimetres divides by 1,000 to give litres, since a litre is defined as exactly 1,000 cubic centimetres. This calculator does that conversion for you.
For a curved solid with a flat circular base, see the cone calculator. For a straight-sided tube, use the cylinder calculator.