Library / Foundations: Maths / FMA-57
Foundations: Maths
Total surface area of a closed cylinder
Whole outside area of a cylinder with both ends closed: the curved wall plus two circular ends.
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- FMA-57
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- v1
- Field
- Foundations: Maths
- Result
- A [m²]
A closed cylinder has a curved wall, which unrolls into a rectangle of width 2πr and height h, and two circular ends of area πr² each. The total is 2πrh + 2πr². For the curved area alone, omit the end terms; for an open-topped tank, include only one end. All lengths must be in the same unit and the area is in that unit squared.
Whole outside area of a cylinder with both ends closed: the curved wall plus two circular ends. For the curved wall alone, leave out the 2πr² term.
Calculator
- Curved area only, 2πrh188.5 m²
- Area of the two ends, 2πr²56.55 m²
- Volume, πr²h282.7 m³
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A cylinder of radius 3 m and height 10 m has a total surface area of 245 m², of which 188.5 m² is the curved wall and 56.55 m² the two ends.
- For a given volume, the surface area is smallest when the height equals the diameter, which is why economical cans are roughly that shape.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| r Radius | 0.1 to 50 m | 6.346 | 18,850 | 18,840 | +10% | +12.5 % |
| h Height | 0.1 to 100 m | 58.43 | 1,942 | 1,883 | +10% | +7.69 % |
Sweep of Radius
| Radius (m) | Total surface area (m²) |
|---|---|
| 0.1 | 6.346 |
| 6.337 | 650.6 |
| 12.57 | 1,784 |
| 18.81 | 3,406 |
| 25.05 | 5,517 |
| 31.29 | 8,117 |
| 37.52 | 11,210 |
| 43.76 | 14,780 |
| 50 | 18,850 |
Sweep one variable
Variables and how each one contributes
Total surface area
What the equation gives you. Everything else feeds it.
Radius
The radius of the circular ends, in metres. Raises the area faster than in proportion, because the radius appears in both the wall and the end-area terms. Raising it by 0.5 m (to 3.5 m) moves the result by +21.2 %, from 245 to 296.9 m².
Height
The perpendicular height of the cylinder between the two ends, in metres. Raises the area in a straight line, because only the curved wall depends on the height. Raising it by 1 m (to 11 m) moves the result by +7.69 %, from 245 to 263.9 m².
Worked example with the values above
- Curved wall: 2πrh = 2π × 3 × 10 = 188.5 m².
- Two ends: 2πr² = 2π × 3² = 56.55 m².
- A = 188.5 + 56.55 = 245 m².
Assumptions and limits
- A right circular cylinder with both ends flat and closed.
- Radius and height in the same length unit; area in that unit squared.
Common mistakes
- Forgetting the two end areas in a closed cylinder.
- Using the diameter in place of the radius.
Statistics and uncertainty
Area scales with the square of a length, so a 1 % error in the radius gives about a 1.5 to 2 % error in the area for a tall cylinder. Measure the diameter carefully when the area drives a coating or heat-loss estimate.
Where it is used
- Paint, insulation or heat-loss area of a tank, drum or pipe with capped ends.
- Material needed to make a closed can.
Maths you need
- Circle area: πr²
- Circumference: 2πr