Sphere Volume Calculator

The volume of a sphere is V = ⁴⁄₃ πr³. Enter a radius above and the calculator returns the volume, the substitution it used, and the same figure in cubic metres, litres and US gallons.

What the formula means

A sphere is the set of all points sitting the same distance from a centre. That distance is the radius, and it is the only measurement a sphere has. Everything else follows from it: the diameter is 2r, the circumference of the widest circle through it is 2πr, the surface area is 4πr², and the volume is four-thirds of π times the radius cubed.

The radius is cubed, which is the part people find counter-intuitive. Double the radius and the volume goes up eight times, not twice, because 2³ = 8. A ball twice as wide holds eight times as much. This is why a small increase in the size of a spherical tank produces a large increase in capacity, and why hail twice the diameter carries eight times the mass.

Where the four-thirds comes from

Archimedes worked it out around 250 BC, without calculus, by comparing a sphere with the cylinder that exactly encloses it. A sphere of radius r fits inside a cylinder of radius r and height 2r. The cylinder's volume is πr² × 2r = 2πr³. Archimedes showed the sphere occupies exactly two-thirds of that, which gives ⁴⁄₃ πr³. He considered the result his finest and asked for a sphere and cylinder to be carved on his tomb.

The modern derivation integrates the area of circular cross-sections along an axis. A slice at height y has radius √(r² − y²), so its area is π(r² − y²). Integrating from −r to r gives π[2r³ − ²⁄₃ r³] = ⁴⁄₃ πr³.

A worked example

Take a radius of 3 units.

• Cube the radius: 3³ = 27
• Multiply by π: 27π
• Multiply by four-thirds: ⁴⁄₃ × 27π = 36π
• As a decimal: 36 × 3.14159265 = 113.0973 cubic units

The exact answer, 36π, is worth keeping. Coursework often asks for the exact form, and it also tells you the decimal is a rounding rather than the true value. The calculator above prints the exact multiple of π whenever the answer lands on one.

If you have the diameter or circumference instead

Most real objects are measured across rather than from the centre, so you usually start with a diameter. Halve it, then use the formula. A ball 20 cm across has a radius of 10 cm and a volume of ⁴⁄₃ π × 1000 = 4188.79 cm³, which is 4.19 litres.

Circumference is the practical one for large objects, because a tape measure goes round something a ruler cannot reach across. Divide the circumference by 2π to recover the radius. A regulation size 7 basketball has a circumference of 0.762 m, giving a radius of 0.1213 m and a volume of 7.47 litres.

Volume against surface area

Volume grows with r³ while surface area grows with r², and the consequences of that gap turn up everywhere. The ratio of surface to volume is 3/r, so it falls as things get bigger.

Small spheres have proportionally enormous surface. This is why fine water droplets in a mist evaporate almost instantly while a puddle takes hours, why crushed ice cools a drink faster than one large cube of the same mass, and why small mammals must eat so much relative to their body weight to replace the heat their surface loses.

Large spheres run the other way. A planet the size of Earth, radius 6371 km, has a volume of about 1.083 × 10¹² km³ and retains internal heat for billions of years, because the surface it loses heat through is tiny compared with the interior generating it.

Units, and the mistake that costs you a factor of a thousand

The formula does not care which unit you use, but it returns the cube of whatever you put in. Radius in centimetres gives cubic centimetres. Radius in metres gives cubic metres. These differ by a factor of a million, so mixing them is the most expensive error available here.

Unit in	Volume out	Useful conversion
centimetres	cm³	1 cm³ = 1 millilitre
metres	m³	1 m³ = 1000 litres
inches	in³	231 in³ = 1 US gallon
feet	ft³	1 ft³ = 7.481 US gallons

The calculator carries the unit through for you and shows litres and gallons alongside the raw figure, which removes the step where most arithmetic goes wrong.

Where it gets used

Storage and pressure vessels. Spherical tanks hold gas under pressure because a sphere has the smallest surface area for a given volume, so it needs the least material and has no corners to concentrate stress. LNG and LPG storage spheres are built this way.

Chemistry and biology. Cell volume, droplet size in an emulsion and the dose held in a spherical capsule all come from this formula. Cell radius matters because nutrients cross the surface while metabolism happens throughout the volume, which places a hard ceiling on how large a cell can grow without folding its membrane.

Sport. Ball specifications are written as circumference because that is what can be measured reliably on a soft object. Converting to volume is how manufacturers check internal pressure and material use.

Astronomy. Planetary densities come from dividing mass by volume, and volume comes from radius. Saturn's density works out below that of water, which is the sort of result you only trust once you have checked the volume calculation.

Halves, caps and shells

A hemisphere is exactly half a sphere, so its volume is ²⁄₃ pi r³. Domes, bowls and the ends of pressure vessels are usually hemispherical, and treating one as a full sphere doubles the answer.

A spherical cap is a slice cut off by a flat plane, smaller than half. Its volume is ^π⁄₃ h²(3r − h), where h is the depth of the cap and r the radius of the original sphere. This is the shape of a shallow dish, or of the liquid sitting in the rounded bottom of a tank, and it is why partially filled spherical vessels are awkward to gauge: the depth and the contents are not proportional.

A hollow shell, such as a ball with a wall thickness, is the difference between two spheres: ⁴⁄₃ π(R³ − r³). For a thin shell this comes close to the surface area multiplied by the thickness, which is a quick way to estimate how much material a spherical vessel needs.

From volume to mass

Volume becomes weight once you multiply by density. A steel ball bearing 10 mm across has a radius of 5 mm, a volume of ⁴⁄₃ π × 125 = 523.6 mm³, and at 7.85 g/cm³ a mass of about 4.11 g. The same ball in aluminium, at 2.70 g/cm³, weighs 1.41 g.

This is the step where unit errors surface. Densities are usually quoted per cubic centimetre while small parts are measured in millimetres, and there are 1000 cubic millimetres in a cubic centimetre.

Why spheres never fill a space

Identical spheres packed as tightly as possible occupy about 74% of their container. Poured in randomly rather than stacked deliberately, they reach roughly 64%.

The practical consequence is that a bin of ball bearings, marbles or spherical catalyst pellets holds noticeably less material than its volume suggests. If you need the mass of a container of spheres, calculate the container volume, multiply by the packing fraction, and only then apply density. Skipping that step overstates the contents by a third.

Common mistakes

Using diameter as radius. This inflates the answer eightfold. If your result looks absurdly large, check this first.

Cubing after multiplying. The exponent applies to r alone, not to ⁴⁄₃ πr. Compute r³ first.

Rounding π too early. Using 3.14 instead of the full value introduces an error of about 0.05%, which is invisible on a homework answer and matters on a tank holding 20,000 litres.

Mixing units mid-problem. A radius in centimetres and a height in metres is the classic way to be out by a factor of a hundred, or a million once cubed.