Volume of a Cone Formula

Geometry Formulas: Area, Volume, Perimeter
Formula:

What is the formula for the volume of a cone?

Answer

Volume of a cone
V = (1/3) · π · r² · h

Here r is the radius, h is the height, perpendicular to the base. The result comes out in cubic units.

Where the formula comes from

V = (1/3) · π · r² · h

Fill a cone with water and pour it into a cylinder of the same base and the same height: it takes exactly three cones to fill it. So the cone is a third of πr²h. The one-third is not an approximation — it is exact for every cone, upright or leaning. (The slant-height relation below, and the surface-area formula, are a different matter: those hold only for a RIGHT circular cone, whose apex sits over the centre of the base.)

Worked examples

Let r = 3, h = 9. Then the volume of a cone ≈ 84.82 cubic units.

Let r = 2.5, h = 6. Then the volume of a cone ≈ 39.27 cubic units.

The common mistake

Using the slant height — the distance along the sloping side — instead of h. The height is the vertical drop from the tip to the centre of the base; the slant is always longer, and the two are linked by l² = r² + h².

Other quantities: Cone

Related Calculations

See Also

Geometry Formulas: Area, Volume, Perimeter

These are the formulas for the area, perimeter, circumference, volume and surface area of the standard shapes — 31 of them. Each page shows not only the formula but where it comes from, what every letter stands for, two worked examples and the mistake that this particular shape invites. For example, it can help you find out what is the formula for the volume of a cone? (The answer is: V = (1/3) · π · r² · h). Pick the pair — "Area of a circle", for instance — and hit the "Find out" button.
Geometry Formulas: Area, Volume, Perimeter
Formula:

Values from the Formula (h = 9)

If r =Volume
1≈ 9.42 cubic units
2≈ 37.7 cubic units
3≈ 84.82 cubic units
4≈ 150.8 cubic units
5≈ 235.62 cubic units
10≈ 942.48 cubic units

FAQ

What is the formula for the volume of a cone?

Volume of a cone V = (1/3) · π · r² · h

Where does this formula come from?

Fill a cone with water and pour it into a cylinder of the same base and the same height: it takes exactly three cones to fill it. So the cone is a third of πr²h. The one-third is not an approximation — it is exact for every cone, upright or leaning. (The slant-height relation below, and the surface-area formula, are a different matter: those hold only for a RIGHT circular cone, whose apex sits over the centre of the base.)

What is the most common mistake here?

Using the slant height — the distance along the sloping side — instead of h. The height is the vertical drop from the tip to the centre of the base; the slant is always longer, and the two are linked by l² = r² + h².

How do I use this formula?

Put your own numbers in place of the letters. With r = 2.5, h = 6, for instance, the formula V = (1/3) · π · r² · h gives ≈ 39.27 cubic units.

See Also