Surface area of a cylinder
The surface area of a cylinder is A = 2πr² + 2πrh, or A = 2πr(r + h): two circular bases of area πr² each, plus a curved side of area 2πrh. A cylinder with a 3 in radius and an 8 in height has 2π × 3 × (3 + 8) ≈ 207.35 in² of surface. The answer is always in square units.
Contents
- The cylinder surface area formula
- Where the formula comes from: unroll the cylinder
- Total, lateral and base areas
- How to find the surface area of a cylinder: worked examples
- Surface area of a cylinder with the diameter
- Cylinder without a top, open tubes and other variants
- Units: why the answer is in square units
- Common mistakes with the area of a cylinder
- How surface area changes with size
- Working backwards: radius or height from the surface area
- Every surface area question, page by page
- FAQ
A = 2πr² + 2πrh. With r = 3 in and h = 8 in: bases 2 × 28.27 = 56.55 in², side 2π × 3 × 8 = 150.80 in², total 207.35 in².
The calculator above takes a radius or a diameter and a height in any unit and shows the total surface area, the curved (lateral) part and each base, with the working. The rest of this guide explains where the cylinder surface area formula comes from, how to adapt it to a cylinder without a lid, and how to avoid the mistakes that cost marks on a test.
The cylinder surface area formula
A closed cylinder has three faces: a circle on top, an identical circle on the bottom, and one curved face that joins them. The surface area is the sum of those three areas.
| Part | Formula | r = 3 in, h = 8 in |
|---|---|---|
| One base (circle) | πr² | 28.27 in² |
| Both bases | 2πr² | 56.55 in² |
| Lateral (curved) surface | 2πrh | 150.80 in² |
| Total surface area | 2πr² + 2πrh = 2πr(r + h) | 207.35 in² |
Here r is the radius of the circular base, h is the height measured straight along the axis, and π is about 3.14159. The factored form 2πr(r + h) is quicker on a calculator: add the radius and the height, multiply by the radius, then by 2π. Both forms give the same number.
Several names point to the same idea. "Area of a cylinder", "cylinder area" and "SA of a cylinder" almost always mean the total surface area. "Curved surface area" and "lateral area" mean the side only. When a question says just "area", check whether the cylinder is closed before you include the two ends.
Where the formula comes from: unroll the cylinder
Cut the top and bottom off a paper tube, then slit the side straight down and flatten it. You get a rectangle and two circles. That flat pattern is the net of a cylinder, and it explains every term of the formula.
- The two circlesEach one has radius r, so each has area πr². Together they give 2πr².
- The rectangle's heightThe side was slit along the axis, so one edge of the rectangle is the height h.
- The rectangle's widthThe other edge used to wrap once around the base, so it equals the circumference of the circle, 2πr.
- Add the piecesThe rectangle's area is width × height = 2πr × h. Total: 2πr² + 2πrh.
The curved face flattens without stretching or tearing, which is why its area is exactly the area of a plain rectangle. That is not true of a sphere: you cannot flatten an orange peel into a rectangle, and the sphere needs a different argument. For a cylinder, the label of a can is a direct demonstration: peel it off and it is a rectangle as wide as the can's circumference and as tall as the can.
Total, lateral and base areas
Most real questions need only part of the surface, so it helps to keep the three pieces separate.
- Lateral surface area, 2πrh. The curved side alone: the label on a can, the paint on a pole, the sheet metal rolled into a duct. The page on the lateral surface area of a cylinder covers it in detail, with examples of labels and poles.
- Base area, πr². One end. It is also the area of the slice you get by cutting the cylinder square to its axis, which is why the cross-sectional area of a cylinder is πr² too, and why the volume is that same area times the height.
- Total surface area, 2πr² + 2πrh. The whole skin of a closed cylinder: the metal of a closed can, the wrapping paper around a round box with a lid.
For a tall, thin cylinder the side dominates. For the 3 in × 8 in example, the lateral area is 150.80 of the 207.35 in², or 8 ÷ 11 of the total. In general, the lateral share is h ÷ (r + h). A flat, wide cylinder such as a coin is the opposite: almost all of its surface is the two faces.
How to find the surface area of a cylinder: worked examples
Each result below was computed with full-precision π and then rounded. Measure both dimensions in the same unit first.
Example 1: a closed can, in inches
A can has a radius of 3 in and a height of 8 in.
- Basesπr² = π × 3² = 28.274 in²; two of them = 56.549 in².
- Side2πrh = 2 × π × 3 × 8 = 150.796 in².
- Total56.549 + 150.796 = 207.345 in², about 207.35 in².
Example 2: a jar, in centimeters
A glass jar is 10 cm across and 15 cm tall, so r = 5 cm. Bases: 2 × π × 5² = 157.08 cm². Side: 2 × π × 5 × 15 = 471.24 cm². Total: 2π × 5 × (5 + 15) = 628.32 cm². That is 200π exactly, a neat check if you work in terms of π.
Example 3: a water tank from its diameter, in feet
A closed tank is 4 ft in diameter and 6 ft tall. You can halve the diameter (r = 2 ft) or use the diameter form below. Bases: 2 × π × 2² = 25.13 ft². Side: 2 × π × 2 × 6 = 75.40 ft². Total: 100.53 ft². If the tank sits on the ground and you only paint the side and the top, leave out one base: 75.40 + 12.57 = 87.96 ft².
Example 4: a drum, in metric
A drum measures 58 cm across and 88 cm tall, so r = 29 cm. One base: π × 29² = 2,642.08 cm². Side: 2 × π × 29 × 88 = 16,034.69 cm². Total: 2 × 2,642.08 + 16,034.69 = 21,318.85 cm², which is 2.1319 m² (divide by 10,000).
Surface area of a cylinder with the diameter
If you measured straight across the end, you have the diameter d = 2r. Substituting r = d ÷ 2 gives:
A = πd²/2 + πdh
The first term is the two bases (each πd²/4) and the second is the side: the circumference πd times the height. For the 4 ft × 6 ft tank: π × 16 ÷ 2 + π × 4 × 6 = 8π + 24π = 32π ≈ 100.53 ft², the same as Example 3. The calculator accepts the diameter directly; the volume side of the same question is covered on the page about the volume of a cylinder with the diameter.
Cylinder without a top, open tubes and other variants
The formula changes whenever a face is missing or extra faces appear. Count the faces that really exist.
| Shape | Faces counted | Formula | Example |
|---|---|---|---|
| Closed cylinder | 2 bases + side | 2πr² + 2πrh | r = 3 in, h = 8 in → 207.35 in² |
| Open top (cup, open tank) | 1 base + side | πr² + 2πrh | r = 4 in, h = 10 in → 301.59 in² |
| Open both ends (tube, pipe) | side only | 2πrh | r = 4 in, h = 10 in → 251.33 in² |
| Half cylinder (closed) | half side + flat face + 2 half-circles | πrh + 2rh + πr² | r = 3 in, h = 8 in → 151.67 in² |
| Hollow cylinder (thick pipe) | outer side + inner side + 2 rings | 2πRh + 2πrh + 2π(R² − r²) | R = 3 in, r = 2 in, h = 8 in → 282.74 in² |
For the open-top cylinder, the radius is 4 in: one base is π × 16 = 50.27 in² and the side is 2π × 4 × 10 = 251.33 in², for a total of 301.59 in². The half cylinder is the shape of a trough or a split log; its volume is on the half cylinder volume page. For the thick pipe, the outer side is 150.80 in², the inner side 100.53 in² and the two ring-shaped ends 31.42 in² together. The bore itself is treated on the hollow cylinder volume page.
Units: why the answer is in square units
Every term of the formula multiplies two lengths: r × r or r × h. A length times a length is an area, so inches give square inches (in²), feet give square feet (ft²), centimeters give cm². Convert with the square of the length factor, not the factor itself.
| From | To | Multiply by |
|---|---|---|
| ft² | in² | 144 (12 × 12) |
| yd² | ft² | 9 (3 × 3) |
| in² | cm² | 6.4516 (2.54 × 2.54) |
| m² | cm² | 10,000 (100 × 100) |
So the 100.53 ft² tank is 100.53 × 144 = 14,476 in², and the 628.32 cm² jar is 628.32 ÷ 6.4516 = 97.39 in². If you mix units, for example a radius in inches and a height in feet, the result is meaningless: convert first.
Common mistakes with the area of a cylinder
- Using the diameter as the radius. The 3 in × 8 in can computed with r = 6 gives 527.79 in² instead of 207.35 in². The base term is four times too big and the side term twice too big.
- Forgetting one base. πr² + 2πrh is right only for an open cylinder. For a closed one you need both ends: 2πr².
- Using the diameter in 2πrh. The side is 2πr × h or πd × h. Writing 2πd × h doubles it.
- Mixing up area and volume. Surface area, 2πr² + 2πrh, is the skin (square units). Volume, πr²h, is the space inside (cubic units). To find the space inside, see how to find the volume of a cylinder.
- Rounding π too early. With 3.14 the can gives 207.24 in² instead of 207.35 in²; acceptable on homework if the question allows it, but keep π in the calculator for real material estimates.
- Squaring the unit at the wrong time. 1 ft² is 144 in², not 12 in².
How surface area changes with size
Scale every length of a cylinder by a factor k and its surface area grows by k². Double both the radius and the height of the 3 in × 8 in can (r = 6 in, h = 16 in) and the surface becomes 829.38 in², exactly four times 207.35 in². The volume grows faster, by k³, so a larger cylinder has less surface for each cubic inch it holds. That is why big tanks lose heat more slowly per gallon than small ones.
For a fixed volume, the closed cylinder with the least surface area has a height equal to its diameter (h = 2r). A 1 liter (1,000 cm³) cylinder needs at least 553.58 cm² of material, with r ≈ 5.42 cm and h ≈ 10.84 cm. The calculus behind this result is shown on the page about deriving the cylinder volume formula.
Working backwards: radius or height from the surface area
If you know the total area A and the height h, the radius solves the quadratic 2πr² + 2πhr − A = 0:
r = (−h + √(h² + 2A ÷ π)) ÷ 2
With A = 300 in² and h = 8 in, r ≈ 3.98 in. If you know A and the radius, the height is simpler: h = A ÷ (2πr) − r. With A = 207.35 in² and r = 3 in, h ≈ 8.00 in, which checks Example 1. Both are built into the radius of a cylinder and height of a cylinder calculators.
Every surface area question, page by page
The surface of a cylinder splits into three topics, each with its own calculator and worked examples.
To go from the surface to the space inside, the cylinder volume calculator on the home page takes the same radius and height and returns the volume in cubic units, gallons and liters.
FAQ
What is the formula for the surface area of a cylinder?
A = 2πr² + 2πrh, which factors to A = 2πr(r + h). The term 2πr² is the area of the two circular bases and 2πrh is the curved side. With r = 3 in and h = 8 in, A = 2π × 3 × 11 ≈ 207.35 in².
How do you find the surface area of a cylinder step by step?
Find the radius (half the diameter). Compute one base, πr², and double it. Compute the curved side, 2πrh. Add the two results. Keep the radius and the height in the same unit; the answer is in square units.
What is the difference between total and lateral surface area?
The lateral surface area is the curved side only, 2πrh. The total surface area adds the two circular ends: 2πrh + 2πr². A label wrapped around a can covers the lateral area; the sheet metal of a closed can covers the total area.
How do you find the surface area of a cylinder with no top?
Count only one base: A = πr² + 2πrh. A cup or an open tank 4 in in radius and 10 in tall has πr² + 2πrh = 16π + 80π = 96π ≈ 301.59 in². A tube open at both ends has only the curved side, 2πrh.
How do I find the radius from the surface area?
Solve 2πr² + 2πhr − A = 0 for r: r = (−h + √(h² + 2A ÷ π)) ÷ 2. With A = 300 in² and h = 8 in, r ≈ 3.98 in. If you know the lateral area L instead, r = L ÷ (2πh).
Is surface area measured in square or cubic units?
Square units: in², ft², cm², m². Volume is the one in cubic units. Multiplying r × r or r × h gives a length times a length, which is an area.
Related
- Lateral areaThe lateral area of a cylinder is L = 2πrh, the curved side only. Calculator, why the formula works, and examples with can labels and painted poles.
- Net of a cylinderA cylinder net is a rectangle 2πr wide and h tall plus two circles of radius r. Draw one to size, print it, and use it to find the surface area.
- How to find the volumeHow to find the volume of a cylinder: square the radius, multiply by π and the height. Worked examples in inches, feet, cm and m, plus common mistakes.
Updated October 12, 2026