How to Find the Volume of a Cone

This is the pyramid formula with a circle underneath it. Base area times height, divided by three, which is what every pointed solid does. The only cone specific part is that the base area happens to be πr², and the only trap is which of the two lengths running up the side counts as the height.

A third of its cylinder

Stand a cone inside the shortest tube that will contain it, matching its base and its height. The cone occupies exactly one third. Fill it three times and the cylinder is full, with nothing spare and no gap at the rim.

That one third is not a property of cones, it is a property of anything that tapers to a point. A pyramid does the same inside its box. Round base or square base, the rule does not notice, which is the strongest argument for learning base area times height over three once rather than learning two formulas that happen to agree.

A worked check you can repeat on paper

That last row is the coincidence flagged on the surface area page, and it is worth seeing from this side too. This cone’s volume and its total surface area both read 301.59. One is cubic units of space, the other square units of material, and they share nothing but digits. It happens only because r is 6 and h is 8, and it is a good reminder that a number matching another number proves nothing at all.

Vertical height, not slant height

A cone gives you two candidate lengths and the two formulas want different ones. Volume wants the vertical height, 8 here. Surface area wants the slant, 10. Swapping them is the most reliable way to get a wrong answer on either page, and neither result looks obviously broken.

If a question hands you the slant and nothing else, recover the height first with Pythagoras. The radius, the height and the slant form a right triangle, so the height is √(l² − r²). Here that is √(100 − 36), which is 8, and only then does the volume formula get to see a number.

Mistakes that catch people out

• Using the slant as the height. Costs 25 percent here, and more on a squat wide cone where the two lengths differ most.
• Forgetting to divide by three. That gives 904.78, the cylinder rather than the cone. Three times too big, and the most common slip on the page.
• Using the diameter as the radius. The radius is squared, so this multiplies the answer by four. A diameter of 12 fed in as a radius gives 1206.37.
• Mixing this up with surface area. Volume divides by three and cubes the units. Surface area does neither. If a 3 has appeared in a surface area calculation, or a slant in a volume, something has crossed over.
• Rounding π before the division. Keep the constant whole and round only the final answer, especially when the result feeds a material order.

Try these

Common questions

What is the formula for the volume of a cone?

π r² h over 3. A cone of radius 6 and height 8 gives π times 36 times 8 over 3, which is 301.59 cubic units, or exactly 96π.

Why divide by three?

Because a cone fills exactly one third of the cylinder that shares its base and height. Here the cylinder holds 904.78 and the cone 301.59. A pyramid does the same inside its box, since the rule belongs to tapering rather than to roundness.

Which height does the volume use?

The vertical one, from the tip straight down to the centre of the base. Using the slant height of 10 instead of the vertical 8 gives 376.99 rather than 301.59, which is 25 percent too much.

How do you find the volume from the slant height?

Recover the vertical height first with Pythagoras, since h = √(l² − r²). A slant of 10 with a radius of 6 gives a height of 8, and the volume follows as normal.

How do you find the height from the volume?

Multiply the volume by three, then divide by the base area. A volume of 301.59 on a base of 113.10 gives 904.78 over 113.10, so the height is 8.

Is a cone the same as a pyramid?

For volume purposes, yes. Both are base area times height divided by three, and the cone formula only looks different because πr² has been substituted in for the base.

Related lessons and calculators

• Surface area of a cone, the same cone measured for material instead of contents, where the slant height is the one that matters.
• Volume of a pyramid, the same one third rule with a flat sided base.
• Volume of a sphere, which completes the cylinder, sphere and cone ratio of 3 to 2 to 1.
• Radius of a cylinder, for working backwards through the solid this one is measured against.
• Volume of 3D shapes calculator, which also handles a cone with its top cut off.
• the National Council of Teachers of Mathematics, for how volume of solids is sequenced in the curriculum.
• Volume of a cylinder, the solid a cone fills exactly one third of.
• Height of a cone, the same formula run backwards when the volume is what you already have.