Volume of a Cone: Formula and Worked Example

The volume of a cone is one-third of the area of its circular base multiplied by its perpendicular height: V = ⅓πr²h. Enter the base radius and height to calculate the cone’s volume in cubic units; the example below uses metres.

Formula

The standard formula is V = πr²h / 3, or equivalently, V = Bh / 3, where:

• V is the cone’s volume, in cubic units such as cubic metres (m³), cubic centimetres (cm³), cubic feet (ft³), or cubic inches (in³).
• r is the radius of the circular base, measured in a unit of length.
• h is the perpendicular height, measured in the same unit as the radius.
• π is pi, approximately 3.14159.
• B is the base area, equal to πr², in square units.

The height is the straight-line distance from the center of the base to the cone’s tip, measured at a right angle to the base. Because the formula squares the radius and multiplies by a length, the result is in cubic units. For example, metres produce cubic metres, not square metres.

Worked Example

Using the calculator’s default radius of 3 m and height of 4 m:

1. Square the radius: 3² = 9 m².
2. Find the base area: π × 9 ≈ 28.2743 m².
3. Multiply by the height: 28.2743 × 4 ≈ 113.0973 m³.
4. Divide by 3: 113.0973 ÷ 3 ≈ 37.70 m³.

The base area is approximately 28.27 m². The cone’s volume is approximately 37.70 m³ when rounded to two decimal places.

Why the Formula Divides by Three

A cone with the same circular base and perpendicular height as a cylinder has one-third of that cylinder’s volume. The cylinder’s volume is πr²h; dividing by three gives the cone formula. This relationship is exact for a right circular cone and also applies to an oblique cone whose base and perpendicular height are the same.

One way to understand the difference is that a cylinder maintains its full circular cross-section from bottom to top, while a cone narrows to a point. Its horizontal cross-sections become smaller as they approach the tip. Adding up those shrinking slices gives one-third of the corresponding cylinder’s volume.

Radius, Diameter, and Slant Height

The radius is the distance from the center of the circular base to its edge. If you are given the diameter, divide it by two before using the formula: r = d/2. Using the diameter as though it were the radius makes the calculated volume four times too large, because the radius is squared.

The slant height is the distance along the cone’s sloping side from the edge of the base to the tip. It is not the height used directly in V = πr²h. For a right circular cone, if you know the slant height ℓ and radius r, you can find the perpendicular height with the Pythagorean theorem:

h = √(ℓ² − r²)

This relationship assumes the cone is a right circular cone, with its tip directly above the center of the base. Once you have h, substitute it and the radius into the volume formula.

Units and Practical Uses

Volume describes three-dimensional space, so its units are always cubed. Keep the radius and height in matching length units before calculating. If one measurement is in centimetres and the other in metres, convert one first; otherwise, the result will be wrong by a large factor.

• For dimensions in centimetres, the result is in cm³. There are 1,000 cm³ in one litre.
• For dimensions in metres, the result is in m³. One cubic metre equals 1,000 litres.
• For dimensions in inches, the result is in in³; dimensions in feet produce ft³.

The formula can help estimate the capacity of a conical container, the amount of material in a cone-shaped pile, or the volume of a geometric model. A real container may have a flat or rounded tip, wall thickness, or a non-circular opening, so a calculation based on an ideal cone may only approximate its usable capacity.

Common Mistakes and Limits

Three errors account for many incorrect answers: using the diameter instead of the radius, using slant height instead of perpendicular height, and forgetting to divide by three. A fourth is mixing units. Check the measurements and units before rounding; rounding intermediate steps too early can also slightly change the final result.

The formula assumes a circular base and a straight-sided cone that narrows to a point. For an oblique cone, the tip is not directly above the base center, but the formula still works if h is the perpendicular distance from the tip to the base plane. It does not use the sloping edge length as height.

For an incomplete cone with its tip cut off parallel to the base, the shape is a conical frustum, not a full cone. Its volume needs both circular radii and the perpendicular distance between the two circular faces. Applying the full-cone formula to that shape will overestimate its volume.

Frequently Asked Questions

What is the formula for the volume of a cone?
Use V = πr²h / 3, where r is the base radius and h is the perpendicular height.

How do you find a cone’s volume from its diameter?
Divide the diameter by two to get the radius, then calculate V = πr²h / 3 using the perpendicular height.

Can I use slant height to calculate cone volume?
Not directly. For a right circular cone, first find the perpendicular height using h = √(ℓ² − r²), where ℓ is slant height.

Why is a cone’s volume one-third of a cylinder’s?
A cone and cylinder with equal base area and perpendicular height have volumes in a 1:3 ratio. The cone’s cross-sections shrink toward its tip.

What units should cone volume use?
Use cubic units: for example, cm³, m³, in³, or ft³. The radius and height must use the same length unit.

Does the formula work for an oblique cone?
Yes. Use the perpendicular height from the tip to the base plane, not the slanted side length.

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in r | Base radius | m | 3 | 0
in h | Height | m | 4 | 0
out volume | Volume | m³ | pi * r^2 * h / 3 | 2
out base_area | Base area | m² | pi * r^2 | 2