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Calculator Answers

Cone Volume Calculator

Source: Euclid, Elements, Book XII, Proposition 10 · Source verified August 20, 2026

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

A cone volume calculator is a geometry tool that computes the 3D space occupied by a cone. It uses the formula V = (1/3)πr²h, requiring the radius of the circular base and the perpendicular height. The resulting volume is expressed in cubic units such as cubic inches, cubic feet, or cubic centimeters, with conversions between them.

Compute the volume of a cone. Enter the base radius and height to find the three-dimensional space inside a cone in cubic units.

See also: Cone Calculator, Volume Calculator – Rectangular Prism, Cylinder, Sphere & Cone, and Pyramid Volume Calculator.

Quick Answer

Find the volume of a cone. Input the radius (or diameter) of the base and the vertical height to calculate volume in cubic units.

Shape

Measurement unit

rh
Dimensions this shape asks for
ft

Measured from the centre of the base to its edge.

ft

That measurement is the

Switch the shape to update the inputs. All dimensions use the same unit; the result also shows the equivalent volume in cubic feet, cubic yards, and cubic metres wherever those differ from it.

Cone volume

Volume

157.08 ft³

V = (1/3) × π × 5² × 6 = 157.08 ft³

FormulaV = (1/3) × π × r² × h
Base area78.5398 sq ft
Cubic yards5.81776
Cubic meters4.448

Steps

  1. Square the radius and multiply by π to get the base area: π × 5² = 78.5398 square feet.
  2. Multiply the base area by the height: 78.5398 × 6 = 471.239 ft³.
  3. Take one third of that: 471.239 ÷ 3 = 157.08 ft³.

This cone holds 157.08 ft³ of space, exactly one third of the 471.239 ft³ cylinder with the same base and perpendicular height.

Volume measures three-dimensional space, so the answer is in cubic units: one ft³ is the space inside a cube measuring one foot on every edge.

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Examples

Radius = 3 ft, Height = 9 ft

Volume ≈ 84.82 cubic feet

Radius = 5 in, Height = 12 in

Volume ≈ 314.16 cubic inches

How it works

Formula · V = (1/3)πr²h · h = √(s² − r²) when only the slant height is known

Volume of a Cone Formula

The volume of a cone is calculated using the base radius and vertical height:

V = (1/3) × π × r² × h

Where:

  • r is the radius of the circular base (half of the diameter).
  • h is the vertical height of the cone.
  • π is Pi (approximately 3.14159).

To calculate the volume of other 3D shapes (such as spheres, cylinders, cubes, or pyramids), check out our comprehensive volume calculator or see our specialized pool volume calculator for swimming pool estimates.

What this page is, and the two neighbours it is not

This page answers one question: how much space is inside a cone. It runs the same engine as the general volume calculator, opened on the cone shape, because a single formula with two inputs does not need a tool of its own to be worth a page of its own.

Two neighbouring pages do different jobs. The cone calculator solves the whole cone rather than just its volume: slant height, base area, lateral surface area and total surface area, from any two of radius, height and slant height. Reach for it when the surface matters, as it does for paint, sheet metal or a paper cone. The volume calculator is the same engine with all six shapes available, for when the cone is one of several things you are measuring.

Where the one third comes from

The fraction is not a fudge factor or a rounded constant. A cone holds exactly one third of the cylinder that shares its base and its height, and exactly is the right word: the ratio is 1 to 3 with nothing left over.

The kitchen demonstration is the one worth remembering. Fill a cone with water and pour it into a cylinder of the same base and height, and it takes three cones to fill it. That is why the formula is the cylinder formula, πr²h, with a third in front of it, and why the third is written as a plain fraction rather than as 0.333.

Cone Volume Calculation Example

Suppose you want to find the volume of a cone with a base radius of 4 cm and a height of 9 cm:

  1. Square the radius: r² = 4 × 4 = 16 cm²
  2. Multiply by height: 16 × 9 = 144
  3. Multiply by π (pi): 144 × 3.14159 ≈ 452.39
  4. Divide by 3 (or multiply by 1/3): 452.39 ÷ 3 ≈ 150.80 cm³

Use the vertical height, not the slant

This is the mistake that costs people the most, because the slant is the easier of the two to measure and the formula wants the other one. The vertical height runs from the centre of the base straight up to the tip. The slant height runs up the outside surface, from the edge of the base to the tip, and it is always the longer of the two.

Using the slant by mistake overstates the volume, and by more than people expect on a wide cone. If the slant is what you can reach, convert it first: the radius, the height and the slant form a right triangle, so h = √(s² − r²). A cone with a radius of 3 and a slant of 5 has a vertical height of 4, not 5, and a volume of 37.70 rather than 47.12.

Cone volumes at a glance

Every row is the formula applied, rounded to two decimals. The units are whatever you measured in, cubed:

RadiusHeightVolume
111.05
2625.13
3984.82
512314.16
615565.49
10101047.20

Edge cases

  • Radius is squared, height is not. Doubling the radius quadruples the volume; doubling the height only doubles it. Getting the two the wrong way round is a large error, not a small one.
  • The slant height is not the height. It is always longer, and using it inflates the answer. Convert with h = √(s² − r²) first.
  • The answer is in cubic units. Three lengths multiplied together give a cubed unit, so centimetres in means cubic centimetres out.
  • Zero and negative values give no volume. A cone with no radius or no height encloses nothing, and a negative length has no meaning.
  • A frustum is not a cone. A bucket or lampshade with the tip cut off needs the whole cone minus the removed tip, not this formula alone.
  • Leaning does not change the volume. An oblique cone with the same base and the same vertical height holds exactly as much as an upright one.

Your measurements stay in the page

The values you enter are processed by this page in your browser. They are not sent to a server, not stored after you close the tab, and not visible to anyone else. There is no account and nothing to sign up for.

Related calculators and resources

Sources

Pi is not cited here: it is a definition rather than a result, and a citation for it would be decoration. The one third is different. That a cone is exactly a third of its cylinder is a theorem, and Euclid records it in Elements, Book XII, Proposition 10: any cone is a third part of the cylinder with the same base and equal height. That is the whole of what the citation supports, and it is what makes the fraction exact rather than a convenient approximation.

Related Calculators

More tools from Education

Cone CalculatorSolve for a cone's height, slant height, base area, lateral area, total surface area, and volume from any two inputs.Volume Calculator – Rectangular Prism, Cylinder, Sphere & ConeCalculate the volume of common 3D shapes including rectangular prism volume, cylinder volume, sphere, cone, and pyramid. Round shapes accept a radius or a diameter. Includes formulas and cubic unit conversions.Pyramid Volume CalculatorCalculate the volume of a square or rectangular pyramid using the base length, base width, and height.Triangular Prism CalculatorCalculate the volume and surface area of a triangular prism with base, height, length, and optional side inputs.

Frequently asked questions

The formula for the volume of a cone is: V = (1/3) × π × r² × h, where 'r' is the radius of the circular base, 'h' is the height of the cone (measured vertically from base to tip), and π (pi) is approximately 3.14159.

A cone has exactly one-third of the volume of a cylinder with the same base radius and height. If you were to fill a cone with water and pour it into a cylinder of equal dimensions, it would take exactly three full cones to fill the cylinder.

Vertical height (h) is the straight line distance from the center of the base to the top point (apex). Slant height (s) is the distance from any point on the base circle's edge to the apex. For volume calculations, always use the vertical height. If you only know the slant height and base radius, you can find the vertical height using the Pythagorean theorem: h = √(s² − r²).

Since the radius is half of the diameter (r = d ÷ 2), first divide the diameter by 2, then use the standard cone volume formula: V = (1/3) × π × (d/2)² × h.

Use this one if volume is the only thing you want. The cone calculator solves the whole cone: slant height, base area, lateral surface area and total surface area as well as volume, from any two of radius, height and slant height. If you need to know how much paint or sheet metal the surface takes, that is the page; if you need how much it holds, this is the shorter route.

Cubic units of whatever you entered. A radius and height in inches give cubic inches; in metres, cubic metres. Because volume is three lengths multiplied together, the unit is cubed rather than carried through unchanged.

Work in centimetres first, because one cubic centimetre is exactly one millilitre. Divide the cubic centimetre figure by 1,000 for litres. Going on to gallons is a further conversion and belongs to a volume converter rather than to this page.

The base has to be a circle, but the tip does not have to sit directly above its centre. A cone leaning to one side has the same volume as an upright one with the same base and the same vertical height. What must be vertical is the height you measure, not the cone itself.

About 84.82 cubic units. Squaring the radius gives 9, multiplying by the height gives 81, multiplying by pi gives 254.47, and dividing by 3 gives 84.82.

It quadruples, because the radius is squared in the formula. Doubling the height only doubles the volume. That asymmetry is why a wide shallow cone holds far more than a narrow tall one of the same overall size.

That is a cone with the tip cut off parallel to the base, like a bucket or a lampshade, and it has a different formula because it has two circular faces rather than one and a point. This page computes the full cone only. For a frustum, work out the whole cone and subtract the small cone you removed.

Yes, exactly rather than approximately. A cone and a cylinder sharing the same circular base and the same height stand in a ratio of exactly 1 to 3. Euclid records the proof, and it is why the one third in the formula is a plain fraction rather than a measured constant.

The calculator will not produce a volume from them. A cone with zero radius or zero height encloses nothing, and a negative length has no meaning, so the tool declines rather than reporting a confident zero or a negative volume.

Because pi does. A cone with whole-number dimensions almost never has a whole-number volume, since the formula multiplies by an irrational constant. Round at the end, to whatever precision your measurements justify, rather than partway through.

Measure the slant height along the outside from the base edge to the tip, and the radius across the base, then use h = √(s² − r²). That is the Pythagorean theorem applied to the right triangle formed by the radius, the height and the slant.