How to use the Sphere Calculator
- Measure or calculate the radius; halve the diameter if that is what you have.
- Enter it in consistent units.
- Read the volume in cubic units of whatever you entered.
- Read the surface area for coatings, insulation or heat-transfer work.
- Use the litres row directly if your radius was in centimetres.
How the calculation works
Archimedes established both sphere formulas over two thousand years ago, and was proud enough of the result to have a sphere inscribed in a cylinder carved on his tomb. The volume is four thirds pi r cubed and the surface area is four pi r squared — exactly four times the area of a great circle through the sphere, and exactly two thirds of the enclosing cylinder's total surface.
The two formulas scale differently, and that difference drives a great deal of physics and biology. Volume grows with the cube of radius while surface area grows only with the square, so the surface-to-volume ratio falls as size increases. Large animals struggle to shed heat, large tanks lose proportionally less through their walls, and small droplets evaporate far faster than big ones — all consequences of that single mismatch in exponents.
Because volume depends on the cube of the radius, measurement error is amplified threefold. A 1% error in radius becomes roughly a 3% error in volume. For tanks and vessels, measure the circumference with a tape and divide by two pi rather than guessing a diameter across a curved surface — the tape measurement is far more repeatable.
V = (4/3)πr³; S = 4πr²Source: Archimedes, On the Sphere and Cylinder (c. 225 BCE); still the standard derivation.
Worked example
A spherical propane vessel measures 1.5 m in radius. What does it hold, and what area needs coating?
- Volume: (4/3) × π × 1.5³ = (4/3) × π × 3.375 = 14.14 m³.
- In litres: 14.14 × 1000 = 14,137 L.
- Surface: 4 × π × 1.5² = 4 × π × 2.25 = 28.27 m².
- At 85% safe fill: 14,137 × 0.85 = 12,016 L usable.
14.14 m³ geometric capacity, about 12,000 L usable at an 85% fill limit, over 28.27 m² of surface to coat.
Frequently asked questions
How do I get the radius from a circumference measurement?+
Divide the circumference by 2π. A 9.42 m tape reading around the equator gives a radius of 1.5 m.
What about a partially filled sphere?+
That needs the spherical cap formula, V = πh²(3r − h)/3, where h is the fill depth. It is not a simple fraction of the total.
Why is surface area exactly four great circles?+
It falls out of Archimedes' cylinder projection, which maps the sphere onto its enclosing cylinder while preserving area exactly.
Does this work for a hemisphere?+
Halve the volume, but for the surface add the flat circular face: 2πr² curved plus πr² flat.
Last reviewed August 31, 2026. We review this page whenever the underlying formula, tax year, published rate or standard changes.