Lateral area of a cylinder
The lateral area of a cylinder is the area of its curved side, without the top and bottom: L = 2πrh, where r is the radius and h the height. With the diameter, L = πdh. A can 3 in in radius and 8 in tall has a lateral surface area of 2 × π × 3 × 8 ≈ 150.80 in².
Contents
L = 2πrh. With r = 3 in and h = 8 in: L = 2 × π × 3 × 8 = 150.80 in².
The calculator above shows the lateral (curved) area as the main reading, with the bases and the total surface next to it. Below: why the formula works, how it differs from the total surface area, and examples with labels, poles and ducts.
Lateral surface area formula of a cylinder
| You know | Formula | Example |
|---|---|---|
| Radius r and height h | L = 2πrh | r = 3 in, h = 8 in → 150.80 in² |
| Diameter d and height h | L = πdh | d = 6 in, h = 8 in → 150.80 in² |
| Circumference C and height h | L = C × h | C = 9.42 in, h = 4.5 in → 42.41 in² |
The three lines are the same formula. The circumference of the base is 2πr, or πd, and the lateral area is simply that circumference times the height. If you can wrap a tape measure around the object, the last form needs no π at all.
The lateral surface is also called the curved surface area. Both names mean the side that a label, a coat of paint or a sheet of insulation would cover.
Why the lateral area is 2πrh
Take a paper tube and cut it straight down the side, parallel to the axis. Flatten it. The curved surface becomes a rectangle, because a cylinder's side bends in only one direction and flattens without stretching.
- Height of the rectangleThe cut ran along the full height, so one side of the rectangle is h.
- Width of the rectangleThe other side used to go once around the base. Its length is the circumference, 2πr.
- AreaWidth × height = 2πr × h = 2πrh.
That rectangle, together with the two circular ends, is the net of a cylinder. The label on a can shows it directly: peel it off carefully and it lies flat as a rectangle, as wide as the can's circumference (plus a small overlap for the glue) and as tall as the can.
There is a second way to see it, with calculus. The volume πr²h grows by about 2πrh × (thin layer) when you add a thin layer to the radius, so the derivative of πr²h with respect to r is exactly 2πrh. The page on how the cylinder volume formula is derived explains this link between the volume and the side.
Lateral and surface area of a cylinder: the difference
The total surface area adds the two bases to the side:
total = 2πrh + 2πr² · lateral = 2πrh · one base = πr²
| Cylinder | Lateral area | Two bases | Total surface area |
|---|---|---|---|
| r = 3 in, h = 8 in | 150.80 in² | 56.55 in² | 207.35 in² |
| r = 5 cm, h = 15 cm | 471.24 cm² | 157.08 cm² | 628.32 cm² |
| r = 2 ft, h = 6 ft | 75.40 ft² | 25.13 ft² | 100.53 ft² |
Which one to use depends on what you are covering. A label, the sides of a tank, a pipe open at both ends, or a tube: lateral area. A closed can or a box with a lid: total area. A cup or an open tank: the side plus one base. The full method for the closed shape is on the surface area of a cylinder page.
The side's share of the total is h ÷ (r + h). For the 3 in × 8 in can that is 8 ÷ 11, about 73 %. The side is larger than one base when h is more than half of r, and larger than both bases together when h is more than r.
How to find the lateral surface area of a cylinder: examples
All results are computed with full-precision π, then rounded.
Example 1: the label of a can (inches)
A can is 3 in across and 4.5 in tall. The label wraps once around it.
- Circumferenceπd = π × 3 = 9.42 in. That is the length of the label strip, before any overlap.
- Lateral areaπdh = 9.42 × 4.5 = 42.41 in².
The same can measured in metric, 7.5 cm across and 11 cm tall: circumference π × 7.5 = 23.56 cm, label area 23.56 × 11 = 259.18 cm².
Example 2: painting a round pole (feet)
A post is 6 in in diameter and stands 10 ft tall. Convert the diameter to feet first: 6 in = 0.5 ft, so r = 0.25 ft.
- Lateral area2πrh = 2 × π × 0.25 × 10 = 15.71 ft².
- Several polesThree columns 10 in across and 9 ft tall: 3 × π × (10 ÷ 12) × 9 = 70.69 ft².
Paint coverage is printed on the can in square feet per gallon; divide the lateral area by that figure and multiply by the number of coats. The top of a post is usually left out, which is exactly why the lateral area, not the total, is the right measure. In metric, a pole 20 cm across and 3 m tall has πdh = π × 0.2 × 3 = 1.885 m².
Example 3: sheet metal for a duct
A round duct 8 in in diameter and 5 ft long needs π × 8 × 60 = 1,507.96 in² of sheet, which is 10.47 ft². Ducts and pipes are open at both ends, so only the curved side counts.
Example 4: radius or height from the lateral area
Turn L = 2πrh around. If L = 150.8 in² and h = 8 in, then r = L ÷ (2πh) = 150.8 ÷ (16π) = 3.00 in. If L = 100 cm² and r = 2 cm, then h = L ÷ (2πr) = 100 ÷ (4π) = 7.96 cm. The radius of a cylinder calculator has a mode for exactly this.
Common mistakes
- Adding the bases when they are not wanted. A label does not cover the top of a can. Use 2πrh alone.
- Using the diameter in 2πrh. 2πdh is twice the real area. Use 2πrh or πdh, never 2πdh.
- Confusing the side with the slice. The flat circle you see when you cut a cylinder straight across has area πr²; that is the cross-sectional area of a cylinder, not its lateral area.
- Mixing inches and feet. In the pole example, 6 in must become 0.5 ft before multiplying by 10 ft.
- Giving the answer in cubic units. Lateral area is an area: in², ft², cm², m². Cubic units belong to the volume, πr²h, which you can get from the cylinder volume calculator.
Quick check: the lateral area is just "distance around × height". If you know the circumference, you do not need π at all.
Lateral area of an oblique cylinder
When the cylinder leans, the formula 2πrh with the vertical height no longer gives the side area. The side of an oblique cylinder with circular bases does not unroll into a simple rectangle. Its area equals the perimeter of a cut made square to the slanted sides times the slant length; that cut is an ellipse, and the perimeter of an ellipse has no exact elementary formula. The volume, on the other hand, still uses the vertical height, as shown on the oblique cylinder volume page. For a right cylinder, which is what textbooks mean by default, L = 2πrh holds exactly.
FAQ
What is the lateral surface area of a cylinder?
It is the area of the curved side only, without the two circular ends. The formula is L = 2πrh, or πdh with the diameter. A cylinder 3 in in radius and 8 in tall has a lateral area of 2π × 3 × 8 ≈ 150.80 in².
What is the difference between lateral and total surface area of a cylinder?
Total surface area adds the two bases to the side: 2πrh + 2πr². Lateral area is the side alone, 2πrh. For the 3 in × 8 in cylinder, the lateral area is 150.80 in² and the total is 207.35 in².
Is curved surface area the same as lateral surface area?
For a cylinder, yes. Both terms mean the rounded side, 2πrh. Some textbooks say curved surface area (CSA) and total surface area (TSA); others say lateral and total.
Why is the lateral surface area 2πrh?
Slit the side along its height and unroll it. It becomes a rectangle whose width is the circumference of the base, 2πr, and whose height is h. The area of that rectangle is 2πr × h.
Can the lateral area be smaller than a base?
Yes, for a short, wide cylinder. The side 2πrh is smaller than one base πr² whenever the height is less than half the radius, as in a coin or a thin disk.
Related
- Surface area of a cylinderThe surface area of a cylinder is A = 2πr² + 2πrh. Get the total, lateral and base areas from a radius or diameter, with worked examples in inches and cm.
- 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.
- Cross-sectional areaThe cross sectional area of a cylinder is πr² for a straight cut. Calculator for straight, slanted and lengthwise cuts, with pipe flow and stress examples.
Updated October 12, 2026