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Formula Reference
Worked example
If we use the result to calculate Radius, we should get the value you entered:
How to calculate the surface area of a cylinder
The surface of a closed cylinder has three parts: two circles (the top and the bottom) and the side. If you unroll the side, it is a rectangle whose width is the circumference of the base and whose height is the height of the cylinder.
\[ \text{Area} = \underbrace{2\pi r^2}_{\text{two bases}} + \underbrace{2\pi r h}_{\text{side}} = 2\pi r (r + h) \]
From the area and the radius you can get the height:
\[ h = \frac{\text{Area}}{2\pi r} - r \]
And from the area and the height, the radius (it comes from a quadratic equation):
\[ r = \frac{-h + \sqrt{h^2 + \frac{2 \times \text{Area}}{\pi}}}{2} \]
How to use the calculator
- Enter two of the three values: the area, the radius and the height.
- Leave empty the one you want to find.
- Press Calculate. If you have the diameter, divide it by 2 to get the radius.
Worked examples
1. Radius of 3 cm and height of 10 cm
\[ \text{Area} = 2\pi \times 3 \times (3 + 10) = 78\pi \approx 245.04\ \text{cm}^2 \]
2. Label for a can
A can has a diameter of 7.5 cm and a height of 11 cm. The label covers only the side: \( 2\pi \times 3.75 \times 11 \approx 259.18\ \text{cm}^2 \). The whole surface, with the lid and the bottom, is \( 2\pi \times 3.75 \times (3.75 + 11) \approx 347.54\ \text{cm}^2 \).
3. Height from the area
A cylinder has a surface area of 300 m² and a radius of 4 m. Its height is \( 300 \div (2\pi \times 4) - 4 \approx 11.94 - 4 = 7.94 \) m.
Open cylinders
- Without a lid (a glass, a tank open at the top): \( A = \pi r^2 + 2\pi r h \).
- Without a top or a bottom (a pipe, a tube): only the side, \( A = 2\pi r h \).
- To paint the outside of a pipe use the outer radius; for the inside, the inner radius.
Surface area of cylinders with a height of 10
| Radius | Two bases | Side | Total area |
|---|---|---|---|
| 1 | 6.28 | 62.83 | 69.12 |
| 2 | 25.13 | 125.66 | 150.8 |
| 3 | 56.55 | 188.5 | 245.04 |
| 4 | 100.53 | 251.33 | 351.86 |
| 5 | 157.08 | 314.16 | 471.24 |
| 6 | 226.19 | 376.99 | 603.19 |
| 8 | 402.12 | 502.65 | 904.78 |
| 10 | 628.32 | 628.32 | 1,256.64 |
For a given volume, the cylinder that uses the least surface is the one whose height equals its diameter. That is why some cans and pots are about as tall as they are wide.
When Do You Need to Calculate the Surface Area of a Cylinder?
Water tanks, silos and round columns are painted or coated by surface. The area tells you how many liters of paint or coating to buy.
Accurate paint and coating quantitiesCan makers calculate the metal sheet for the body and the lids, and the label covers the side. Small savings per can add up over millions of units.
Material per unit in packaging designInsulation for pipes, ducts and water heaters is sold by surface. The side area of the pipe gives the amount of insulation, and heat loss depends on it.
Size insulation and estimate heat lossWrapping a cylindrical box, a candle or a bottle needs paper for the side and both ends. The surface area gives the minimum size of paper.
Cut wrapping paper with less wasteHeat exchangers, reactors and test tubes exchange heat through their surface. Engineers use the surface area to size them.
Key for heat-transfer calculationsSurface area problems with cylinders combine the area of a circle and of a rectangle, and are common in school exams.
Practice with π, circles and nets of solidsCommon Mistakes
⚠️ Forgetting the Two Bases
⚠️ Using the Diameter as the Radius
⚠️ Counting the Wrong Number of Bases
⚠️ Confusing Area and Volume
⚠️ Height and Radius in Different Units
⚠️ Ignoring Overlaps
Applications by Industry
Manufacturing & Packaging
- Cans: Metal sheet for bodies and lids
- Labels: Label size from the side surface
- Tubes: Material for cardboard tubes and containers
Energy & HVAC
- Insulation: Insulation for pipes, ducts and water heaters
- Heat exchangers: Transfer surface of the tubes
- Boilers: Heat loss through the shell
Painting & Coatings
- Tanks: Paint for water tanks and silos
- Columns: Coating for round columns and posts
- Pipelines: Anti-corrosion coating per meter of pipe
Chemistry & Labs
- Reactors: Wall area for cooling jackets
- Test tubes: Glass surface in contact with the sample
- Batteries: Casing area of cylindrical cells
Construction
- Formwork: Cylindrical formwork for round columns
- Waterproofing: Membrane for round cisterns
- Cladding: Sheet metal for round towers
Education
- Geometry: Nets of solids and surface area
- Optimization: The can that uses the least material for a given volume
- Exams: Problems mixing the area and the volume of cylinders
External Resources
- Cylinder surface area and volume — Math is Fun
- Cylinder definition, properties and surface area — Wikipedia