Hong Life CalculatorsUnderstanding the cylinder calculator — concept and principle
A right circular cylinder is the shape of a can, a pipe or a round tank: two equal circles joined by a curved side that meets them at right angles. From the radius r and the height h, this calculator gives the volume, the lateral (curved side) area and the total surface area. It is used to find how much liquid a tank or pipe holds, how much sheet metal or label material a can needs, and how much paint covers a column.
How it is calculated
- Volume: V = πr²h
- Lateral area: L = 2πrh
- Total surface area: T = 2πr(r + h) = 2πrh + 2πr²
r and h must use the same unit. The volume is in that unit cubed and the areas in that unit squared. If you know the diameter, use r = d ÷ 2. For capacity, 1,000 cm³ = 1 litre and 1 m³ = 1,000 litres.
The principle
Every horizontal slice of a cylinder is the same circle of area πr². Stacking that area through the height h gives the volume, base area × height, the same rule as for any prism. Cavalieri’s principle extends this to a slanted (oblique) cylinder: if every slice has the same area, the volume is the same, as long as h is the perpendicular height. For the surface, cut the curved side along a straight line and unroll it. It becomes a rectangle whose width is the circumference 2πr and whose height is h, so its area is 2πrh. Adding the two circular ends, 2 × πr², gives the total.
Worked example
- Inputs: r = 3, h = 10.
- Base area: π × 3² = 9π ≈ 28.274334.
- Volume: 9π × 10 = 90π ≈ 282.7433388.
- Lateral area: 2 × π × 3 × 10 = 60π ≈ 188.4955592.
- Total surface: 2 × π × 3 × (3 + 10) = 78π ≈ 245.044227.
- If the units are centimetres, the cylinder holds about 282.7 cm³, or 0.283 litres.
| Radius | Volume |
|---|---|
| 0.75 | 17.67145868 |
| 1.5 | 70.68583471 |
| 2.25 | 159.0431281 |
| 3 | 282.7433388 |
| 3.75 | 441.7864669 |
| 4.5 | 636.1725124 |
| 6 | 1,130.973355 |
How to read the result
For a fixed volume, a closed cylinder uses the least material when its height equals its diameter (h = 2r). Many cans are taller than this for reasons of handling, stacking and labelling. Volume scales with r² but only linearly with h, so a small increase in diameter adds far more capacity than the same increase in height.
Common mistakes
- Entering the diameter as the radius, which makes the volume four times too large.
- Forgetting one or both circular ends when the total surface is needed, or including them when only the side (a label or a pipe wall) matters.
- Using the slanted length of an oblique cylinder instead of the perpendicular height.
Frequently asked questions
How do I find the height needed for a given volume?
h = V ÷ (πr²). To hold 1 litre (1,000 cm³) with r = 5 cm, h ≈ 12.73 cm.
How much water does a pipe hold per metre?
Use the inner radius: πr² × 1 m. A pipe with an inner diameter of 10 cm holds about 7.85 litres per metre.
What about a hollow cylinder (tube)?
Its material volume is π(R² − r²)h, where R and r are the outer and inner radii.
Sources: Archimedes, On the Sphere and Cylinder, Book I · B. Cavalieri, Geometria indivisibilibus continuorum (1635)