The volume of a cone is V = (1/3)πr²h, where r is the base radius and h is the perpendicular height. A cone holds exactly one third of a cylinder with the same base and height. A cone of radius 3 m and height 2 m holds (1/3) × π × 9 × 2 = 6π ≈ 18.85 m³.
Volume of a cone formula
V = (1/3)πr²h
- r = radius of the circular base (half the diameter)
- h = perpendicular height, from the apex straight down to the centre of the base
- π ≈ 3.14159
If you know the diameter d instead, V = πd²h / 12. The answer comes out in cubic units of whatever length you used: m³ if r and h are in metres, cm³ if they are in centimetres.
Why is there a one third in the cone formula?
The cylinder comparison
Fill a cone with water or sand and pour it into a cylinder with the same base and height. It takes three cone-fulls to fill the cylinder. The cylinder holds πr²h, so the cone holds (1/3)πr²h. This is a demonstration, not a proof, but it matches the result exactly.
Derivation by integration
Slice the cone into thin circular discs parallel to the base. Measure y downwards from the apex. By similar triangles, the disc radius at depth y is ry/h, so the disc area is π(ry/h)² and its volume is π(ry/h)² dy. Adding all discs from y = 0 to y = h:
V = ∫₀ʰ π r² y² / h² dy = (πr² / h²) × (h³ / 3) = (1/3)πr²h
The one third comes from integrating y², which is why every pyramid (square, triangular, any base) also has V = (1/3) × base area × height.
Slant height and the cone
The slant height l runs from the apex down the sloping side to the edge of the base. The radius, height and slant height form a right triangle, so by the Pythagorean theorem:
l = √(r² + h²) and so h = √(l² − r²)
Slant height is used for the curved surface area (πrl). For volume you always need the perpendicular height h, so convert first if a question gives you l.
Volume of a frustum (truncated cone)
Cut the top off a cone parallel to the base and you get a frustum, the shape of a bucket, a hopper or a lampshade. With bottom radius R, top radius r and perpendicular height h:
V = (1/3)πh(R² + Rr + r²)
Setting r = 0 gives back the cone formula, and setting r = R gives the cylinder πR²h, which is a quick check that you have written it correctly.
Worked examples
Example 1: volume of a sand heap
A conical heap of sand on site has a base diameter of 6 m and a height of 2 m. Find its volume.
r = 6 / 2 = 3 m, h = 2 m.
V = (1/3) × π × 3² × 2 = (1/3) × π × 18 = 6π ≈ 18.85 m³.
For comparison, a cylinder of the same size would hold 18π ≈ 56.55 m³, exactly three times as much.
Example 2: given the slant height
A cone has base radius 5 cm and slant height 13 cm. Find its volume in cm³ and litres.
h = √(13² − 5²) = √(169 − 25) = √144 = 12 cm.
V = (1/3) × π × 25 × 12 = 100π ≈ 314.16 cm³ = 0.314 litres (1 litre = 1000 cm³).
Example 3: conical hopper (frustum)
A grain hopper is a frustum with a top diameter of 3 m, an outlet diameter of 0.6 m and a height of 2 m. Find its capacity.
R = 1.5 m, r = 0.3 m, h = 2 m.
R² + Rr + r² = 2.25 + 0.45 + 0.09 = 2.79 m²
V = (1/3) × π × 2 × 2.79 = 1.86π ≈ 5.84 m³, which is about 5,843 litres.
Unit conversions for cone volume
| From | To | Multiply by |
|---|---|---|
| m³ | litres | 1,000 |
| m³ | cm³ | 1,000,000 |
| litres | cm³ (mL) | 1,000 |
| cm³ | m³ | 0.000001 |
| mm³ | cm³ | 0.001 |
A cube of 1 m side is 100 cm on each edge, so it holds 100 × 100 × 100 = 1,000,000 cm³, not 100 cm³. Convert lengths to one unit before you calculate, not after.
Cone and related solids compared
| Solid | Volume | r = 3 m, h = 2 m |
|---|---|---|
| Cone | (1/3)πr²h | 18.85 m³ |
| Cylinder | πr²h | 56.55 m³ |
| Hemisphere (radius r) | (2/3)πr³ | 56.55 m³ |
| Frustum | (1/3)πh(R² + Rr + r²) | depends on top radius |
Common mistakes
- Using the diameter as the radius: this makes the answer four times too big, because r is squared.
- Using the slant height as h: the volume needs the perpendicular height. Find h = √(l² − r²) first.
- Forgetting the 1/3: gives the cylinder volume, three times too big.
- Mixed units: a radius in cm and a height in m gives nonsense. Convert first.
- Wrong litre conversion: 1 m³ is 1,000 litres, and 1 litre is 1,000 cm³.
Where the cone volume formula is used
- Construction: estimating stockpiles of sand, aggregate and soil. For slabs, footings and columns, a concrete volume calculator handles the common shapes.
- Process and storage: hoppers, silo bottoms and funnels are cones or frustums.
- Manufacturing: material in turned conical parts, nozzles and countersinks.
- Everyday: ice-cream cones, traffic cones and paper cups.
FAQs
What is the formula for the volume of a cone?
V = (1/3)πr²h, where r is the base radius and h is the perpendicular height. The result is in cubic units, such as m³ or cm³.
Why is the volume of a cone one third of a cylinder?
Slicing the cone into thin discs and integrating gives πr²h/3, because the disc area grows with the square of the distance from the apex. Three cone-fulls of water exactly fill a cylinder of the same base and height.
How do you find the volume of a cone with slant height?
First find the perpendicular height using h = √(l² − r²), then use V = (1/3)πr²h. For r = 5 cm and l = 13 cm, h = 12 cm and V ≈ 314.16 cm³.
What is the volume of a frustum of a cone?
V = (1/3)πh(R² + Rr + r²), where R and r are the two end radii and h is the perpendicular height between them.
How do I convert cone volume from m³ to litres?
Multiply by 1,000. A cone of 18.85 m³ holds 18,850 litres.