Geometry
How To Calculate The Surface Area Of A Cylinder
A cylinder is the easiest curved solid to understand, because unrolling its side turns it into a plain rectangle whose width is the circle's circumference.
Quick Answer
surface area = 2 x pi x r x h + 2 x pi x r^2
- A
- Total surface area including both ends
- r
- Radius of the circular base
- h
- Height of the cylinder
The curved side is two pi r h, and the two flat ends add another two pi r squared. A cylinder of radius three and height ten has a curved area of 188.4956 and a total surface area of 245.0442.
What Is The Surface Area Of A Cylinder?
The surface area of a cylinder is the curved side plus the two circular ends. A cylinder of radius three and height ten has a curved area of 188.4956, two ends totalling 56.5487, and a total surface area of 245.0442.
The curved part is the one that surprises people, and unrolling it explains why. Cut the side vertically and flatten it, and you get a rectangle whose height is the cylinder's height and whose width is the circumference of the circle, two pi r. Multiply the two and you have the curved area.
That is why the curved area formula is two pi r h. The circumference is two pi r and the height is h, so the rectangle is that wide and that tall. There is no approximation involved; the unrolled side really is a rectangle.
The ends are ordinary circles, each with an area of pi r squared. There are two of them, so they contribute two pi r squared between them. For a radius of three that is 56.5487, a little under a quarter of the total.
A tube with open ends is a different calculation. If only the curved surface counts, the answer is two pi r h on its own. Paint on the outside of a pipe, or the label wrapped around a can, is exactly this quantity.
Surface area and volume are separate formulas and are constantly confused. Volume is pi r squared h, the base area times the height, and it is measured in cubic units. Surface area is measured in square units and answers a different question about material rather than capacity.
This is the formula behind everyday manufacturing. The metal in a tin can is its surface area, the label is the curved area, and the amount of soup is the volume. Minimising the surface area for a given volume is exactly the problem that makes drinks cans the proportions they are.
A can is most material-efficient when its height equals its diameter. That is a genuine optimisation result, and it is why so many tins look similar, though marketing and handling push real products away from the ideal.
Units behave as for any area. A radius in centimetres and a height in centimetres give a surface area in square centimetres. If one is in metres and the other in centimetres, convert first, or the answer will be out by a factor of a hundred.
The radius is half the diameter, and it is the radius that gets squared and multiplied by pi. Using the diameter by mistake quadruples the end area and doubles the curved area, which produces an answer that is wrong but not obviously so.
Finally, the same reasoning covers half a cylinder, or a cylinder cut lengthwise. Split the shape into its flat faces and its curved face, work out each one and add. Almost every surface area problem reduces to that habit.
Formula
A = 2 x pi x r x h
The unrolled side, which is a rectangle of width two pi r and height h.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| A | Curved area | square metres | Area of the side only, without the ends. |
| r | Radius | metres | Radius of the circular base. |
| h | Height | metres | Height of the cylinder. |
A = 2 x pi x r x h + 2 x pi x r^2
The curved side plus the two circular ends.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| A | Total area | square metres | Side plus both ends. |
| r | Radius | metres | Radius of the base. |
| h | Height | metres | Height of the cylinder. |
V = pi x r^2 x h
Base area times height. A different question from surface area, in cubic rather than square units.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| V | Volume | cubic metres | Space inside the cylinder. |
| r | Radius | metres | Radius of the base. |
| h | Height | metres | Height of the cylinder. |
How To Calculate The Surface Area Of A Cylinder
- 1
Find the radius and the height
Halve the diameter if that is what you measured. Both measurements must be in the same unit before anything is multiplied.
- 2
Work out the circumference of the base
Two pi r, which for a radius of three is about 18.85. This becomes the width of the rectangle you get by unrolling the side.
- 3
Multiply by the height for the curved area
18.85 times ten is 188.4956. This is the side alone, which is what you need for a label or for painting a pipe.
- 4
Add the two circular ends
Each end is pi r squared, about 28.2743, so the pair adds 56.5487. The running total becomes 245.0442.
- 5
Decide whether the ends belong
An open tube has no ends, so the answer is the curved area alone. A closed can needs both, and a can with one open end needs just one.
Examples
Example 1: A cylinder of radius three and height ten
- Radius
- 3
- Height
- 10
| Step | Calculation | Result |
|---|---|---|
| Base area | pi x 3^2 | 28.2743 |
| Curved area | 2 x pi x 3 x 10 | 188.4956 |
| Total surface area | 188.4956 + 56.5487 | 245.0442 |
Result: The cylinder covers 245.0442 square units in total, of which 188.4956 is the curved side and 56.5487 is the two ends.
Example 2: A storage tank of radius five and height twelve
- Radius
- 5 m
- Height
- 12 m
| Step | Calculation | Result |
|---|---|---|
| Base area | pi x 5^2 | 78.5398 |
| Curved area | 2 x pi x 5 x 12 | 376.9911 |
| Total surface area | 376.9911 + 157.0796 | 534.0708 |
Result: A tank of radius five and height twelve needs 534.0708 square metres of steel, and the ends account for 157.0796 of that.
Example 3: A label wrapped around a tin of radius four
- Radius
- 4 cm
- Height
- 7 cm
| Step | Calculation | Result |
|---|---|---|
| Base area | pi x 4^2 | 50.2655 |
| End area for both ends | 2 x 50.2655 | 100.531 |
| Curved area | 2 x pi x 4 x 7 | 175.9292 |
Result: The label on a tin of radius four and height seven covers 175.9292 square centimetres, because only the curved side is printed.
Calculator
Total surface area
245.0442
- Curved surface area
- 188.4956
- Area of both ends
- 56.5487
- Volume
- 282.7433
Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.
Prefer a full-width tool? Open the The Surface Area Of A Cylinder calculator page.
Common Mistakes
Using the diameter instead of the radius
The radius is half the diameter and it appears twice in the end area, so the error compounds. Using the diameter makes the ends four times too large and the side twice too large.
Forgetting that there are two ends
A closed cylinder has two circles, not one. Counting only one understates the total, and on a short wide cylinder that is a large fraction of the answer.
Including the ends on an open tube
A pipe or a label has no ends, so the correct figure is the curved area alone. Adding ends that do not exist overstates the material needed.
Confusing surface area with volume
Surface area is square units and volume is cubic. Both start with pi and the radius squared, which makes them easy to mix up, but they answer different questions.
Mixing units between radius and height
A radius in centimetres with a height in metres gives a meaningless result. Convert both to the same unit first.
Using the height where the circumference belongs
The curved area is circumference times height, not height squared. Writing two pi h squared is a common slip that produces a plausible but wrong number.
Assuming a taller cylinder always has more area
A very short wide cylinder can have more surface area than a tall thin one. The two terms trade off, which is exactly the tension in can design.
FAQ
What is the formula for the surface area of a cylinder?
Two pi r h for the curved side, plus two pi r squared for the two circular ends. Add the two parts for the total.
Why is the curved area two pi r h?
Because unrolling the side gives a rectangle. Its width is the circumference of the base, two pi r, and its height is the cylinder's height, so the area is the product.
How do you find the surface area of an open tube?
Use the curved area alone, two pi r h. There are no ends to add, which is why a pipe or a can label needs less material than a sealed can.
How is surface area different from volume?
Surface area is the outside skin, in square units, and answers how much material is needed. Volume is pi r squared h, in cubic units, and answers how much the cylinder holds.
What shape of can uses the least material?
For a given volume, the most efficient cylinder has its height equal to its diameter. Real cans deviate for handling, shelf appearance and manufacturing reasons.
Does the formula work for a half cylinder?
Split it into faces. Half the curved area plus the flat rectangular cut plus the two half circle ends, worked out separately and added.
References
- [1]Wikipedia, Cylinder — https://en.wikipedia.org/wiki/Cylinder
- [2]Wikipedia, Surface area — https://en.wikipedia.org/wiki/Surface_area
- [3]Math is Fun, Cylinder — https://www.mathsisfun.com/geometry/cylinder.html
- [4]Wikipedia, Surface area of a cylinder — https://en.wikipedia.org/wiki/Cylinder#Surface_area