Volume of a Cone: Formula, Worked Examples and Why ⅓

The volume of a cone is one third of the area of its circular base multiplied by its height: V = ⅓πr²h, where r is the radius of the base and h is the perpendicular height from the base to the tip. A cone with a radius of 3 cm and a height of 4 cm holds ⅓ × π × 3² × 4 = 12π ≈ 37.7 cm³, exactly a third of the cylinder that would fit round it.

What is the volume of a cone formula?

The volume of a cone formula is:

V = ⅓ π r² h

Symbol	Meaning	Units
V	volume	cm³, m³ (cubed units)
r	radius of the circular base	cm, m
h	perpendicular height, from the centre of the base straight up to the apex	cm, m
π	pi, about 3.142	no units

The apex is the pointed tip. The height must be the straight-up height, not the length along the sloping side (that is the slant height, l).

How to find the volume of a cone, step by step

1. Find the radius. If you are given the diameter, halve it.
2. Square the radius: r².
3. Multiply by π to get the area of the base: πr².
4. Multiply by the height: πr²h (this would be the volume of a cylinder).
5. Divide by 3.
6. Give the answer in cubed units, rounded as the question asks.

Worked example 1

A cone has a radius of 5 cm and a height of 12 cm. Find its volume to 1 decimal place.

V = ⅓ × π × 5² × 12 = ⅓ × π × 25 × 12 = 100π = 314.2 cm³

Worked example 2: given the diameter

An ice-cream cone is 6 cm across the top and 10 cm deep. How much ice cream fits inside (level with the top)?

radius = 6 ÷ 2 = 3 cm

V = ⅓ × π × 3² × 10 = 30π = 94.2 cm³

That is about 94 ml, because 1 cm³ = 1 ml.

Why is the volume of a cone a third of a cylinder?

The volume of a cone is a third of a cylinder with the same base and height because a cone gets narrower all the way to a point, so it holds much less near the top. If you slice a cylinder horizontally, every slice is the same circle. Slice a cone and the circles shrink steadily from r at the base to nothing at the apex. Adding up all those shrinking slices (which calculus does exactly) gives exactly one third.

You can check it with water: fill a cone and pour it into a cylinder with the same radius and height. It takes three cones to fill it. The same rule works for a pyramid: a pyramid holds a third of the prism around it, V = ⅓ × base area × height. Archimedes went further about 2,200 years ago and showed that a cone and a sphere of the same radius, poured together, exactly fill the cylinder round them.

Cone volume with slant height

If a question gives the slant height l instead of the height, use Pythagoras' theorem first. The radius, the height and the slant height make a right-angled triangle inside the cone, with l as the hypotenuse:

h = √(l² − r²)

Worked example 3

A cone has a radius of 6 cm and a slant height of 10 cm. Find its volume.

h = √(10² − 6²) = √(100 − 36) = √64 = 8 cm

V = ⅓ × π × 6² × 8 = 96π = 301.6 cm³

What is the surface area of a cone?

The surface area of a cone is the area of its circular base plus its curved surface: A = πr² + πrl. When you unroll the curved surface it becomes a sector of a circle whose radius is the slant height l, with an area of πrl.

For the cone in example 3: A = π × 6² + π × 6 × 10 = 36π + 60π = 96π ≈ 301.6 cm². (Here the surface area and the volume happen to have the same number; their units are different.)

Measurement	Formula	Uses
Volume	⅓πr²h	perpendicular height h
Curved surface area	πrl	slant height l
Total surface area	πr² + πrl	both the base and the curved surface
Slant height	√(r² + h²)	Pythagoras

Common mistakes

• Using the slant height in the volume formula. Volume needs the perpendicular height; find it with Pythagoras first.
• Forgetting to divide by 3. Without the ⅓ you have the volume of a cylinder.
• Using the diameter instead of the radius. Halve it first, then square it.
• Squaring the wrong thing. Only the radius is squared: ⅓π × r² × h, not ⅓π × (rh)².
• Wrong units. Volume is in cm³ (cubed); surface area is in cm² (squared).

Frequently asked questions

What is the formula for the volume of a cone?

The formula is V = ⅓πr²h, where r is the radius of the circular base and h is the perpendicular height from the base to the apex. It is one third of the volume of a cylinder with the same radius and height. For example, a cone with radius 2 cm and height 9 cm has a volume of ⅓ × π × 4 × 9 = 12π ≈ 37.7 cm³.

How do you find the volume of a cone without the height?

If you know the slant height l and the radius r, find the height with Pythagoras' theorem: h = √(l² − r²), because r, h and l form a right-angled triangle inside the cone. Then use V = ⅓πr²h. If you know the volume and the radius instead, rearrange the formula: h = 3V ÷ (πr²).

Why do you divide by 3 for a cone?

A cone narrows to a point, so its horizontal slices shrink from a full circle at the base to nothing at the tip. Adding up all those slices gives exactly one third of the cylinder with the same base and height. You can test it by filling a cone with water three times and pouring it into the matching cylinder: it fills it exactly.

How do you find the radius of a cone from its volume?

Rearrange V = ⅓πr²h to make r the subject: r = √(3V ÷ (πh)). For example, a cone holding 150 cm³ with a height of 8 cm has r = √(450 ÷ 25.13) = √17.9 ≈ 4.2 cm.

Try it yourself

Open the 3D Shapes & Nets tool, choose Cone and switch to Volume. Press Fill with water and watch the level speed up near the narrow top, then turn on Slice it and move the plane up to see the circular cross-sections shrink to a point. Switch back to Net and unfold the cone to see its curved surface flatten into a sector. Volume links to density too: see the Chemistry Lab, where liquids are measured in cm³.