Sphere Volume Calculator
Two thirds of its cylinder — Archimedes' favourite result.
Work out Sphere Volume. Two thirds of its cylinder — Archimedes' favourite result. Exact where an exact definition exists.
Volume (cubic units)
113.097336
Surface area 113.0973
A sphere fills 52.36% of the smallest cube containing it, which is π/6 regardless of size. Archimedes considered the relationship between a sphere and its circumscribing cylinder — volume exactly two thirds — his finest result, and had it carved on his tomb.
How the Sphere Volume Calculator works
Volume and surface area of a sphere, with the great circle area and the share of the enclosing cube. That share is π/6 for every sphere, and the relationship to its circumscribing cylinder is exactly two thirds.
Also known as: volume of a sphere · ball volume calculator · four thirds pi r cubed · sphere surface area calculator
Archimedes and the cylinder
A sphere's volume is exactly two thirds of the smallest cylinder containing it, and its surface area is exactly two thirds of that cylinder's total surface too. Both relationships hold precisely, for every sphere.
Archimedes considered this his finest result and asked for the sphere-and-cylinder figure to be carved on his tomb. Cicero reported finding the tomb, overgrown, by that carving nearly two centuries later.
The surface area being exactly four times the great circle area is the other surprise. Four πr² against πr² is not obvious from any picture, and it is what makes the sphere formulas so tidy.
The square-cube law
Surface area scales with the square of radius and volume with the cube, so the surface-to-volume ratio falls as size increases. Doubling the radius multiplies area by four and volume by eight.
That single relationship explains a great deal of biology. Small animals lose heat faster relative to their mass and so need proportionally more food; cells stay small because nutrients enter through a surface and are used by a volume.
It governs engineering too. A larger reactor is harder to cool, a larger structure is proportionally heavier, and a scale model does not behave like the full-size object because the ratios have changed.
Packing, and why bubbles are round
A sphere fills π/6 of its enclosing cube — 52.36%. The densest possible packing of equal spheres fills about 74% of space, a fact conjectured by Kepler and proved only in recent decades.
Random packing manages about 64%, which is why a jar of marbles has so much air in it and why granular materials settle when shaken. The shaking moves the arrangement towards the denser ordered state.
Bubbles are spherical because a sphere has the least surface area for a given volume, and surface tension minimises area. Any other shape has more surface and therefore more energy, so it relaxes towards the sphere.
Where to go next
The Sphere Volume question rarely arrives on its own. These are the ones that usually come with it:
- Cylinder Volume Calculator — Widening beats heightening — radius is squared.
- Cone Volume Calculator — Exactly one third of its cylinder, always.
- Circle Area Calculator — A circle fills 78.54% of its square, always.
- Percentage Calculator — Every common percentage question in one place.
Frequently asked questions
What is the formula for sphere volume?
Four thirds πr³. Volume scales with the cube of radius, so doubling the radius multiplies volume by eight.
What is the surface area of a sphere?
4πr², which is exactly four times the area of its great circle. That relationship was one of Archimedes' results and it is not obvious.
Why did Archimedes want a sphere on his tomb?
He considered the relationship between a sphere and its circumscribing cylinder — volume exactly two thirds, surface area also exactly two thirds — his finest work, and asked for the figure to be carved on his tomb.
How much of a cube does a sphere fill?
52.36%, which is π/6, regardless of size. It is why packing spheres wastes so much space, and why the densest possible packing still leaves about 26% empty.
How do I find the radius from the volume?
Multiply by 3, divide by 4π, then take the cube root. A volume of 113.1 gives a radius of 3.
Why do bubbles form spheres?
Because a sphere has the smallest surface area for a given volume, and surface tension minimises area. Any other shape has more surface and therefore more energy.
What is the volume of a spherical cap?
πh²(3r − h) ÷ 3, where h is the cap height. It is the calculation for liquid in a spherical tank or the volume of a dome.
How efficiently do spheres pack?
The densest possible arrangement fills about 74% of space, proved only in recent decades. Random packing manages around 64%, which is why a jar of marbles has so much air in it.
What is a hemisphere's surface area?
Half the sphere plus the flat circular face: 2πr² + πr² = 3πr². Forgetting the flat face is the usual error.
Why do planets form spheres?
Because gravity pulls equally in all directions and a sphere is the shape that minimises gravitational potential energy. Bodies below a certain size lack the gravity to overcome their own strength and stay irregular.
How does surface area scale with volume?
Area with the square of radius and volume with the cube, so the surface-to-volume ratio falls as size increases. It is why small animals lose heat faster and why cells stay small.
What is the volume of the Earth?
About 1.08 trillion cubic kilometres, from a mean radius of roughly 6,371 km. The Earth is very slightly flattened at the poles, so a sphere is a close approximation rather than exact.
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