Volume Of a Cone Calculator

What this calculator does

A cone has a circular base that tapers up to a single point. An ice cream cone, a traffic cone, a funnel, a party hat. This works out its volume, the space inside, from the radius of its base and its height.

Type the two in, pick a unit, and you have the volume.

Using the calculator

1. Type the base radius, from the centre of the circular base to its edge.
2. Type the height, straight up from the base to the tip.
3. Pick the unit, then press Calculate.

Both values have to be positive, and the volume comes back in cubic units of the unit you choose.

The formula | volume = (π × radius² × height) ÷ 3

The volume of a cone is:

volume = (π × radius² × height) ÷ 3

It is the cylinder's formula with a divide by three on the end. The base is a circle of area π × radius², and a straight cylinder on that base and height would hold π × radius² × height. A cone narrows to a point instead of staying full to the top, so it holds exactly a third of that. The next section is the nicest way to see why.

Why a cone is a third of a cylinder

Here is the fact worth remembering: a cone is exactly one third of the cylinder with the same base and the same height.

You can check it with water. Take a cone-shaped cup and a cylinder of matching base and height, fill the cone, and pour it in. It takes precisely three conefuls to fill the cylinder. That one third is the whole difference between the two formulas, and it is why the cone's has a divide by three where the cylinder's does not.

The height, not the slant

The height the formula wants is the straight, vertical distance from the centre of the base up to the tip. It is not the slanted side that runs from the rim to the tip. That longer slanted line is the cone's "slant height", and it is used for surface area, not for volume. For volume, measure straight up.

The value of π it uses

The tool sets π to 3.141592654, ten significant figures, which is far finer than anything you could measure on a real cone. So π is never the weak link, and your answer is only as accurate as the radius and height you put in. Round the result back to match how carefully you measured those.

A worked example | radius 10 cm, height 20 cm

Say the radius is 10 cm and the height is 20 cm.

1. Area of the circular base: π × 10² = 314.159 cm².
2. Times the height: 314.159 × 20 = 6,283.19.
3. Divide by 3: 6,283.19 ÷ 3 = 2,094.4 cm³.

So the volume is about 2,094 cm³. Notice the middle step, 6,283, is the volume of the matching cylinder, and the cone comes to exactly a third of it.

The cone and the pyramid

A cone is really a pyramid with a round base, and a pyramid is a cone with a flat, straight-sided base. They obey the very same rule: each is a third of the straight solid, prism or cylinder, that shares its base and height. For the flat-based version, see the volume of a pyramid calculator.

Questions people ask

What is the volume of a cone with radius 10 and height 20?

About 2,094 cm³. Find the base area π × 10², multiply by the height 20, and divide by 3.

Why do you divide by 3?

Because a cone is one third of the cylinder with the same base and height. Three conefuls fill that cylinder exactly.

Is it the base area times the height?

Yes, times one third. The base is a circle of area π × radius², multiplied by the height and then by a third.

Is the height the slanted side?

No. The height is the straight vertical distance from the base to the tip. The slanted side is the slant height, used for surface area, not volume.

What value of π does it use?

It uses π as 3.141592654, ten significant figures, finer than any real measurement.

References

A note on where this comes from. That a cone is one third of its cylinder, like a pyramid to its prism, follows from Cavalieri's principle, that solids with equal cross-sections at every height have equal volume. The base area π × radius² rests on the value of π tabulated by the US National Institute of Standards and Technology. For further reading, see Cone.

1. Cavalieri's principle, the basis for a cone being one third of the cylinder that shares its base and height.
2. National Institute of Standards and Technology (NIST), Digital Library of Mathematical Functions, value of π used in the circular base area. https://dlmf.nist.gov/