Volume of Cones: Formula, Derivation & Worked Examples

A cone is a circular base tapering to a single point — an ice-cream cone, a funnel or a traffic cone. Its volume is exactly one-third of the volume of the cylinder with the same base and height: V = (1/3)πr²h. This is not a coincidence; it is one of the most useful facts in solid geometry, and it can be checked with a simple experiment.

The one-third rule matters because conical containers are everywhere. A conical measuring cup holds exactly one-third of what its cylindrical twin would hold, and grain heaps poured from a truck naturally form cones. Knowing the derivation — three cones of water fill the matching cylinder — turns the formula from a memorised fact into something obvious.

Key Concepts

• Radius (r): the radius of the circular base of the cone.
• Vertical height (h): the perpendicular distance from the vertex to the centre of the base — not the slant height.
• Matching cylinder: a cylinder with the same base radius r and the same height h; its volume is πr²h.
• The one-third factor: pouring a cone full of sand into the matching cylinder fills it exactly one-third of the way; three cones fill it completely.
• Volume is measured in cubic units (cm³, m³).

Formulas

• Volume of a cone: V = (1/3)πr²h.
• Derivation idea: the cone and the matching cylinder share the base area πr²; the cone's cross-sections shrink toward the tip, and the total works out to one-third of the cylinder's volume.
• Finding height from volume: h = 3V ÷ (πr²).
• The formula assumes a right circular cone whose vertex sits directly above the centre of the base.

Worked Example

Find the volume of a cone with radius 6 cm and vertical height 7 cm. Use π = 22 ÷ 7.

1. Write the formula: V = (1/3) × π × r² × h = (1/3) × (22 ÷ 7) × 36 × 7.
2. Cancel the 7s: (22 ÷ 7) × 7 = 22, so V = (1/3) × 22 × 36.
3. Compute 22 × 36 = 792; then 792 ÷ 3 = 264. The volume is 264 cm³.
4. Check against the matching cylinder: πr²h = 22 × 36 = 792 cm³, and 792 ÷ 3 = 264, which confirms the one-third rule.
5. In litres this is 264 ÷ 1000 = 0.264 litres.

Common Misconceptions

• Forgetting the 1/3: πr²h alone is the volume of the cylinder, not the cone; the cone is one-third of that.
• Using the slant height l as h: volume needs the vertical height, the perpendicular drop from the tip to the base centre.
• Thinking the rule works for mismatched pairs: the one-third relationship holds only when the cone and cylinder share both the base radius and the height.

Real-World Applications

• Designing funnels, conical flasks and measuring cups.
• Estimating the volume of grain heaps, sand piles and conical stockpiles.
• Sizing conical tents and party hats.
• Calculating the capacity of conical hoppers in factories.

Three cones fill the matching cylinder, so a cone's volume is (1/3)πr²h. Use the vertical height rather than the slant height, remember the one-third factor every time, and check your answer by comparing it with the cylinder that shares the same base and height.