Volume Of Sphere Calculator

What this calculator does

A sphere is a perfectly round ball, every point on its surface the same distance from the centre. A marble, a globe, a planet, a bubble. Because it is so even, a single number fixes its size, the radius, and this works out the volume from it.

Type the radius, pick a unit, and you have the space inside.

Using the calculator

1. Type the radius, from the centre of the sphere to its surface.
2. Pick its unit.
3. Press Calculate.

The radius has to be positive, and the volume comes back in cubic units of that unit. If you only have the diameter, the width straight across, halve it first to get the radius.

The formula | volume = (4 × π × radius³) ÷ 3

The volume of a sphere is:

volume = (4 × π × radius³) ÷ 3

The radius is cubed, raised to the third power, because volume lives in three dimensions. That is multiplied by π and by four thirds. The one thing to be careful of is to use the radius, not the diameter. A sphere described by its width across needs that halved first.

Archimedes and the two thirds

This formula carries one of the oldest celebrated results in mathematics. Around 250 BCE, Archimedes proved that a sphere takes up exactly two thirds of the smallest cylinder that fits around it, the cylinder with the same radius and a height of two radii. He was prouder of this than of anything else he did, and asked for a sphere sitting inside a cylinder to be carved on his tombstone.

The bounding cylinder holds 2 × π × radius³, and two thirds of that is (4 × π × radius³) ÷ 3, exactly the sphere's volume. There is even a neat link to the cone next door: a hemisphere is what is left when you scoop a cone out of a cylinder, so the very same one third that shrinks a cone is what builds the four thirds in the sphere. See the cylinder calculator for the shape it is measured against.

Why the volume grows so fast

Because the radius is cubed, a sphere's volume climbs steeply as it gets bigger. Double the radius and the volume goes up not twice but eight times, since 2 cubed is 8. A ball twice as wide holds eight times as much. It is why a small change in radius makes such a large change in how much a round thing can hold.

The value of π it uses

The tool carries π to ten significant figures, 3.141592654. That is more precise than any radius you could realistically measure, so π is not what limits the answer. The radius is. Once you have the result, round it to match how carefully the radius was measured.

A worked example | radius 10 cm

Say the radius is 10 cm.

1. Cube the radius: 10³ = 1,000.
2. Multiply by 4 and by π: 4 × π × 1,000 = 12,566.37.
3. Divide by 3: 12,566.37 ÷ 3 = 4,188.79 cm³.

So the volume is about 4,189 cm³. Bump the radius up to 20 cm and the volume would not double, it would jump eightfold, to roughly 33,500 cm³.

Questions people ask

What is the volume of a sphere with radius 10?

About 4,189 cm³. Cube the radius, multiply by 4 and by π, then divide by 3.

Why is it four thirds?

Because a sphere is two thirds of the cylinder that just contains it, as Archimedes proved, and two thirds of that cylinder's 2π × radius³ works out to four thirds π × radius³.

Do I use the radius or the diameter?

The radius. If you have the diameter, the distance straight across, halve it first.

What happens to the volume if I double the radius?

It grows eight times, not two, because the radius is cubed and 2 cubed is 8.

What value of π does it use?

It uses π as 3.141592654, ten significant figures.

References

A note on where this comes from. That a sphere is two thirds of its circumscribing cylinder, with the sphere's volume following as four thirds π × radius³, was proved by Archimedes in On the Sphere and Cylinder. The value of π used is the one tabulated by the US National Institute of Standards and Technology. For further reading, see Sphere.

1. Archimedes, On the Sphere and Cylinder (c. 225 BCE), proving a sphere is two thirds the volume of its circumscribing cylinder.
2. National Institute of Standards and Technology (NIST), Digital Library of Mathematical Functions, value of π. https://dlmf.nist.gov/