Core answer: Cone volume V = (1/3)πr²h — exactly one third of the cylinder with the same base and height. Lateral (side) surface = πrl where l = √(r² + h²) is the slant height. A traffic cone, an ice-cream cone, and a funnel all obey these two formulas.
The formula set
| Quantity | Formula | Example (r=3, h=4) |
|---|---|---|
| Slant height | l = √(r² + h²) | 5 (a 3-4-5!) |
| Volume | V = πr²h/3 | 37.7 |
| Lateral surface | S = πrl | 47.1 |
| Total surface | πr(r + l) | 75.4 |
Why exactly one third?
Archimedes showed (and calculus confirms) a cone fills precisely 1/3 of its circumscribing cylinder. Practical consequence: a conical cup holds a third of what the same-height glass does — and the classic gelateria trick is that a "generous" tall cone often holds less than a short cup.
Worked examples
Example 1 — Ice cream honesty. Cone r = 3 cm, h = 12 cm: V = π×9×12/3 ≈ 113 cm³ ≈ 113 mL. A half-scoop ball on top (r = 3 cm hemisphere): (2/3)π×27 ≈ 57 mL. Total ≈ 170 mL — compare with the cup price.
Example 2 — Sand pile. A conical sand pile 2 m high with 3 m diameter base (r = 1.5): V = π×2.25×2/3 ≈ 4.7 m³. At ~1.6 t/m³ that's ~7.5 tonnes — one small truckload.
Example 3 — Party hat from a sheet. A hat of slant l = 20 cm and base r = 8 cm needs a sector of radius 20 cm; the sector arc must equal the base circle 2π×8 ≈ 50.3 cm, so the sector angle = 50.3/(2π×20) × 360° ≈ 144°.
Common mistakes and myths
- Using vertical height in πrl — lateral surface needs the SLANT height l = √(r²+h²); using h undercounts.
- Forgetting the 1/3 — πr²h is the cylinder; the cone is one third of it.
- Including the base when you shouldn't — a funnel has no base; a party hat has no base; total vs lateral surface matters.
- Thinking taller always means more volume — at fixed slant height, the maximum-volume cone has h = l/√3; tall skinny cones hold surprisingly little.
- Mixing up cone and pyramid — same 1/3-base×height rule, but the base area formula differs (πr² vs polygon area).