The volume and surface area formulas
V = 1/3 πr²h
A = πr² + πrl
Where the formula comes from
The one-third factor is the same one that governs every pyramid, and it holds for any base shape: a cone is just a pyramid with a circular base. The slant height comes from Pythagoras, l = √(r² + h²), because the radius, the height and the slant form a right triangle. The lateral surface πrl is what you get when you unroll the cone into a circular sector.
A worked example, step by step
A cone with a base radius of 3 cm and a height of 4 cm
- Base area: π × 3² = 28.27 cm².
- Volume: 1/3 × 28.27 × 4 = 37.70 cm³.
- Slant height: √(3² + 4²) = √25 = 5 cm.
- Lateral surface: π × 3 × 5 = 47.12 cm².
Watch the units: volume goes in cubic units and surface area in square ones. Mixing them is the most common mistake.
Where it turns up in practice
Hoppers, funnels and piles of loose material, which naturally settle into a cone at their angle of repose. Also traffic cones, loudspeaker drivers and, of course, ice cream — where the one-third rule explains why a cone holds so much less than it looks like it should.