Core answer: Cylinder volume V = πr²h; lateral surface S = 2πrh; total surface = 2πr(r + h). A standard 500 mL bottle (r ≈ 3.3 cm, h ≈ 15 cm) checks out: π×10.9×15 ≈ 513 cm³. Cans are optimized so that at minimum material, height equals diameter — most food cans nearly obey this.
The formula set
| Quantity | Formula | Example (r=4 cm, h=10 cm) |
|---|---|---|
| Volume | πr²h | 502.7 cm³ |
| Lateral surface | 2πrh | 251.3 cm² |
| Total surface | 2πr² + 2πrh | 351.9 cm² |
Worked examples
Example 1 — Water heater capacity. An 80 L tank: inner diameter 40 cm (r = 20 cm), so height = 80,000/(π×400) ≈ 63.7 cm. Check your bathroom fit before ordering.
Example 2 — Pipe flow. Doubling pipe diameter from DN50 to DN100: cross-section = πr² goes from 19.6 to 78.5 cm² — 4× the flow capacity at the same pressure, not 2×.
Example 3 — Label printing. A can r = 3.25 cm, h = 11.5 cm: label size = circumference 2π×3.25 ≈ 20.4 cm × height 11.5 cm, plus ~5 mm overlap.
Example 4 — Well volume. A well of diameter 1 m, water depth 6 m: V = π×0.25×6 ≈ 4.7 m³ ≈ 4,700 L.
The optimal can
For fixed volume, minimum surface area occurs at h = 2r (height = diameter). Real cans deviate for stacking, grip, and shelf optics — a "sleek" 330 mL can uses ~8% more aluminum than the optimum, paid for as design.
Common mistakes and myths
- Confusing lateral and total surface — paint on a closed tank needs total (2 caps + side); a label needs lateral only.
- Using diameter as radius — r = d/2; a "10 cm pipe" has r = 5 cm. Error quadruples the volume.
- Assuming bigger tank = proportionally bigger surface — volume grows with r²h while surface grows slower; big tanks lose proportionally less heat per liter.
- Ignoring wall thickness — capacity uses INNER radius; strength calculations use outer. A 2 mm wall on a 200 mm tank removes ~2% of capacity.
- Thinking any h/r ratio stores the same — at fixed volume, tall-thin cylinders cost more material than the h = 2r optimum.