Enter the radius of the base and the height of the cylinder. The surface area (S = 2πr(r + h)) is shown with its parts (base area and lateral area), along with conversions between cm² and m².
Use the same unit for both and enter numbers only (no units. For 3 cm, enter "3"). If you only know the diameter, divide it by 2 to get the radius first.
Result and figure
Enter the radius of the base and the height in the fields on the left and press "Calculate". The result will appear here.
Table of Contents
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What you can do on this page
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What is this calculation used for?
-
How to Use
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Formulas and figures
-
Symbols and terms
-
Good to know before you start
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How to calculate it in Excel
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How to calculate it in Google Sheets
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How to calculate it in Python
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How to write it in LaTeX and other math languages (copy and paste)
-
How to have ChatGPT do the calculation
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DataChef Features
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Related Features
-
NumberChef Calculators List
What you can do on this page
- Enter the radius of the base and the height, and the surface area of the cylinder (\(S = 2\pi r(r+h)\)) is calculated on the spot
- The result also breaks the total down into the base area (\(\pi r^2\)), the two bases together and the lateral area (\(2\pi rh\)), so you can see where each part comes from
- The result also shows the area in another unit: square feet (ft²) if you enter inches, and square inches (in²) if you enter feet. Switch "Units" to Metric to use cm² and m² instead
- The result is also drawn as a 3D shape. Turn it with the mouse or a swipe to check its shape
- A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page
Enter the radius and the height in the same unit (both in inches, or both in feet). For how much a cylinder holds (its volume), see the cylinder volume calculator page.
What is this calculation used for?
Estimating the metal in a drink can (manufacturing and design)
A 12-ounce soda can is close to a cylinder with a radius of 1.3 in and a height of 4.8 in. Its surface area is \(2\pi \times 1.3 \times (1.3 + 4.8) \approx 49.8\,\mathrm{in^2}\), and the metal in one can is roughly this area.
Billions of cans are made every year, so choosing a shape with less surface area for the same volume (the right balance of radius and height) saves a lot of material. This formula is a basic tool in can design.
Paint for a tank or a drum (construction and facilities)
A 55-gallon drum, with a radius of about 11.25 in and a height of about 33.5 in, has a surface area of \(2\pi \times 11.25 \times (11.25 + 33.5) \approx 3163\,\mathrm{in^2}\), or about 22 ft².
Paint cans list how many square feet they cover, so once you know the surface area you can work out how much to buy. For rust-proofing and waterproofing jobs too, the estimate starts with the surface area.
Estimating pipe insulation (plumbing and HVAC)
To wrap a pipe with an outside diameter of 2 in (radius 1 in, or 1/12 ft) that runs 30 ft, the insulation covers the lateral area, \(2\pi \times \tfrac{1}{12} \times 30 \approx 15.7\,\mathrm{ft^2}\) (the ends of the pipe connect to other pipes, so only the side counts).
Jobs such as insulating hot water pipes or preventing condensation use this lateral area formula to estimate materials. It is a classic case of using only the part of the surface area you need.
The acetate collar or chocolate band around a cake (baking)
An 8-inch round cake (radius 4 in) that is 4 in tall has a lateral area of \(2\pi \times 4 \times 4 \approx 100.5\,\mathrm{in^2}\), and a collar wrapped around its side must be as long as the circumference, \(2\pi \times 4 \approx 25.1\) in.
Thinking of the side as a rectangle (width = circumference, height = height), just as on this page, tells you how much acetate or chocolate you need for the decoration.
Estimating fabric for a homemade bottle cover (crafts)
To make a cover for a water bottle with a radius of 1.5 in and a height of 8 in, the fabric for the side is a rectangle about 9.4 in wide (the circumference) and 8 in tall, with an area of \(2\pi \times 1.5 \times 8 \approx 75.4\,\mathrm{in^2}\). The bottom needs one circle, about \(\pi \times 1.5^2 \approx 7.1\,\mathrm{in^2}\) (add seam allowances in practice).
For round pouches, pencil cup covers and anything else shaped like a cylinder, the idea of the net goes straight into making the pattern.
Formulas and figures
Surface area of a cylinder (the basic idea)
Figure
Standard notation
(the usual math form)
\(S\)
\(=\)
\(B\)
\(\times\)
\(2\)
\(+\)
\(L\)
In words
(symbols replaced with words)
④
\(S\): surface area
\(=\)
①
\(B\): base area (one circle)
\(\times\)
②
\(2\): number of bases
\(+\)
③
\(L\): lateral area
The formula in words
①
Take the
\(B\): base area (one circle)
②
multiply it by the
\(2\): number of bases
for the top and the bottom
③
add the
\(L\): lateral area
④
and you get the
\(S\): surface area
Quick example
The surface area of a cylinder with a base radius of 3 in and a height of 4 in is
\(S\): surface area
\(=\)
base area (about 28.27 in²)
\(\times\)
number of bases (2)
\(+\)
lateral area (about 75.40 in²)
\(\pi \times 3^2 = 9\pi, \quad 9\pi \times 2 = 18\pi\)
\(2 \times \pi \times 3 \times 4 = 24\pi\)
\(18\pi + 24\pi = 42\pi \approx 131.9\,\mathrm{in^2}\)
Key idea
The basic way to think about surface area is "the total area of the net" (the solid cut open and laid flat). The net of a cylinder is 2 circles of the same size (the top and bottom bases) plus 1 rectangle (the side), so the surface area is base area × 2 + lateral area. US textbooks often write this as \(S = 2B + L\).
Unlike volume (\(\pi r^2 h\)), the answer multiplies two lengths, so its unit is squared: enter inches and you get square inches (in²); enter feet and you get square feet (ft²).
Base area (the area of the base circle)
Figure
Standard notation
(the usual math form)
\(B\)
\(=\)
\(\pi\)
\(\times\)
\(r\)
\(2\)
In words
(symbols replaced with words)
④
\(B\): base area
\(=\)
①
\(\pi\): pi
\(\times\)
②
\(r\): radius of the base
③
squared (the number times itself)
The formula in words
①
Take
\(\pi\): pi
②
and the
\(r\): radius of the base
③
multiply pi by the radius
squared (the number times itself)
④
and you get the
\(B\): base area
Quick example
The base area (one circle) of a cylinder with a base radius of 3 in is
\(B\): base area
\(=\)
\(\pi\): pi
\(\times\)
radius of the base (3 in)
squared
\(\pi \times 3^2 = 9\pi \approx 28.27\,\mathrm{in^2}\)
Key idea
This is the same as the formula for the area of a circle, \(\pi\) × radius × radius. A cylinder has two bases of the same size, one on top and one on the bottom, so for the surface area this is doubled. Only the radius is squared.
Lateral area (the side unrolls into a rectangle)
Figure
Standard notation
(the usual math form)
\(L\)
\(=\)
\(2\)
\(\times\)
\(\pi\)
\(\times\)
\(r\)
\(\times\)
\(h\)
In words
(symbols replaced with words)
⑤
\(L\): lateral area
\(=\)
②
\(2\): turns the radius into the diameter
\(\times\)
③
\(\pi\): pi
\(\times\)
①
\(r\): radius of the base
\(\times\)
④
\(h\): height
The formula in words
①
Take the
\(r\): radius of the base
②
multiply it by
\(2\): turns the radius into the diameter
③
multiply by
\(\pi\): pi
to get the circumference of the base (\(2\pi r\))
④
multiply that by the
\(h\): height
⑤
and you get the
\(L\): lateral area
Quick example
The lateral area of a cylinder with a base radius of 3 in and a height of 4 in is
\(L\): lateral area
\(=\)
radius to diameter (2)
\(\times\)
\(\pi\): pi
\(\times\)
radius of the base (3 in)
\(\times\)
height (4 in)
\(2 \times \pi \times 3 \times 4 = 24\pi \approx 75.40\,\mathrm{in^2}\)
Key idea
Cut the side of a cylinder straight down and unroll it, and you get one rectangle. The width of this rectangle is the circumference of the base (\(2\pi r\)), and its height is the height of the cylinder (\(h\)).
So the lateral area is just the area of that rectangle, width × height: circumference × height = \(2\pi r \times h\).
The combined formula (all in one step)
Standard notation
(the usual math form)
\(S\)
\(=\)
\(2\)
\(\times\)
\(\pi\)
\(\times\)
\(r\)
\(\times\)
\((r+h)\)
In words
(symbols replaced with words)
⑤
\(S\): surface area
\(=\)
①
\(2\): turns the radius into the diameter
\(\times\)
②
\(\pi\): pi
\(\times\)
③
\(r\): radius of the base
\(\times\)
④
\((r+h)\): radius plus height
The formula in words
①
Multiply together
\(2\): turns the radius into the diameter
②
\(\pi\): pi
③
\(r\): radius of the base
④
and
\((r+h)\): radius plus height
⑤
and you get the
\(S\): surface area
Quick example
The surface area of a cylinder with a base radius of 3 in and a height of 4 in, in one step, is
\(S\): surface area
\(=\)
radius to diameter (2)
\(\times\)
\(\pi\): pi
\(\times\)
radius of the base (3 in)
\(\times\)
radius plus height (3 + 4 = 7 in)
\(2 \times \pi \times 3 \times (3+4) = 42\pi \approx 131.9\,\mathrm{in^2}\)
Key idea
Put formulas 2 and 3 into formula 1 and you get \(S = 2\pi r^2 + 2\pi rh\). Both terms contain \(2\pi r\); factoring it out gives \(S = 2\pi r(r+h)\).
Use formulas 1 to 3 to understand where the area comes from, and this combined formula to calculate it in one go on a calculator or in Excel. The answer is of course the same (\(42\pi\) in this example).
Converting area units (in² to ft²)
Figure
Standard notation
(the usual math form)
\(S_{\mathrm{ft^2}}\)
\(=\)
\(S_{\mathrm{in^2}}\)
\(\div\)
\(144\)
In words
(symbols replaced with words)
③
\(S_{\mathrm{ft^2}}\): surface area in ft²
\(=\)
①
\(S_{\mathrm{in^2}}\): surface area in in²
\(\div\)
②
\(144\): square inches in 1 ft²
The formula in words
①
Take the
\(S_{\mathrm{in^2}}\): surface area in in²
②
divide it by
\(144\): square inches in 1 ft²
③
and you get the
\(S_{\mathrm{ft^2}}\): surface area in ft²
Quick example
A 55-gallon drum (a cylinder with a radius of about 11.25 in and a height of about 33.5 in) has a surface area of about 3163 in². In square feet, that is
surface area in ft²
\(=\)
surface area in in² (3163)
\(\div\)
square inches in 1 ft² (144)
\(3163 \div 144 \approx 21.97\,\mathrm{ft^2}\)
Key idea
The conversion factor for area is the length factor squared. Since 1 ft = 12 in, \(1\,\mathrm{ft^2}\) is \(12 \times 12 = 144\,\mathrm{in^2}\). A common mistake is to divide by 12 instead of 144.
To go back from square feet to square inches, multiply by 144. Do not mix this up with the factor for volume (\(1\,\mathrm{ft^3} = 12^3 = 1728\,\mathrm{in^3}\), the cube).
The surface area of a cylinder is "base area (πr²) × 2 + lateral area (2πrh)". The key is that the side unrolls into a rectangle whose width is the circumference and whose height is the height of the cylinder. Combined, S = 2πr(r + h). The answer is in the square of the unit you entered (inches give in²).
Symbols and terms
Symbols
| \(S\) | ess | A common symbol for area. On this page it stands for the surface area of the cylinder (the total area of all its faces). |
| \(B\) | bee | The base area (the area of one base circle), from the first letter of "base". It is found with the circle area formula \(\pi r^2\). |
| \(L\) | capital L | The lateral area (the area of the whole side), from the first letter of "lateral". It is circumference × height = \(2\pi rh\). |
| \(r\) | ar | The radius of the base circle (the distance from its center to its edge), from the first letter of "radius". It is half the diameter. |
| \(h\) | aitch | The height of the cylinder (the straight distance from the bottom face to the top face), from the first letter of "height". |
| \(\pi\) | pi | The Greek letter for the ratio of a circle's circumference to its diameter. It goes on forever as \(3.14159\ldots\), and 3.14 is a common rounded value. |
| \(r^2\) | r squared | The number \(r\) multiplied by itself (\(r \times r\)). The small raised 2 is an exponent that shows how many times the number is used as a factor. |
| \(2\pi r\) | two pi r | The circumference of the base circle (the distance around it): diameter (\(2r\)) × pi. It is the width of the rectangle you get when you unroll the side. |
| \(\mathrm{in^2}\) | square inches | A unit of area. A square that is 1 inch on each side has an area of 1 in². |
| \(\mathrm{ft^2}\) | square feet | A unit of area. A square that is 1 foot on each side has an area of 1 ft². \(1\,\mathrm{ft^2} = 144\,\mathrm{in^2}\). |
Terms
| cylinder | A straight solid with circular bases, where every slice across it is a circle of the same size. Cans, drums and paper towel tubes are all around us. |
| surface area | The total area of the outside of a solid. For a cylinder, it is the two bases (top and bottom) plus the side. It tells you how much material covers the surface, such as wrapping or paint. |
| base | The top and bottom faces of a prism or cylinder. A cylinder has two circles of the same size, one at the top and one at the bottom, and both are called bases. |
| base area | The area of a base. For a cylinder the base is a circle, so the base area comes from the circle area formula \(\pi r^2\) (\(\pi\) × radius × radius). |
| lateral surface | The side of a solid, all the way around. The side of a cylinder is one curved surface that becomes a rectangle when you cut it straight down and unroll it. |
| lateral area | The area of the whole side. For a cylinder, it is the area of the unrolled rectangle: circumference × height = \(2\pi rh\). |
| net | The flat shape you get by cutting a solid open and laying it out. The net of a cylinder is 2 circles plus 1 rectangle, and the surface area is the total area of the net. |
| circumference | The distance around a circle: diameter × pi = \(2\pi r\). The width of the unrolled side of a cylinder is exactly the circumference. |
| pi | The number of times a circle's diameter fits around its circumference. It is the same for every circle, is written \(\pi\), and is about 3.14. |
| factoring out a common factor | Pulling a factor that every term shares out in front of parentheses. Taking the common \(2\pi r\) out of \(2\pi r^2 + 2\pi rh\) gives \(2\pi r(r+h)\). |
| unit conversion | Writing the same amount in a different unit. For area, the key point is that the area factor is the length factor squared (1 ft = 12 in, so 1 ft² = 144 in²). |
Good to know before you start
Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
| Area and circumference of a circle (Grade 7) |
|
| Area of a rectangle (Grades 3–4) |
|
| Nets (Grades 6–7) |
|
| Units of area (Grades 3–5) |
|
| Pi and algebraic expressions (Grades 7–8) |
|
How to calculate it in Excel
Copy the whole table below and paste it into cell A1 in Excel. It works as is.
Table to find the surface area of a cylinder (two bases + side)
| Radius of the base r (in) | 3 |
| Height h (in) | 4 |
| Both bases, 2πr² (in²) | =2*PI()*B1^2 |
| Lateral area, 2πrh (in²) | =2*PI()*B1*B2 |
| Surface area (in²) | =B3+B4 |
Table to find the base area (one circle)
| Radius of the base r (in) | 3 |
| Base area, πr² (in²) | =PI()*B1^2 |
Table to find the lateral area
| Radius of the base r (in) | 3 |
| Height h (in) | 4 |
| Lateral area, 2πrh (in²) | =2*PI()*B1*B2 |
Table to find it with the combined formula 2πr(r + h)
| Radius of the base r (in) | 3 |
| Height h (in) | 4 |
| Surface area, 2πr(r+h) (in²) | =2*PI()*B1*(B1+B2) |
Table to convert square inches to square feet
| Surface area in in² | 3163 |
| Surface area in ft² | =B1/144 |
After pasting, column A holds the labels and column B holds the numbers. The upper rows are your inputs, and the formulas in the lower rows calculate automatically from them.
In a formula, "PI()" is the Excel function that returns pi (3.14159…), "B1" and "B2" mean "use the number in that cell", "*" is multiplication, "/" is division and "^" is an exponent (how many times to multiply).
In the first table, for example, B3 shows about 56.549, B4 about 75.398 and B5 about 131.947 (in², since the inputs are in inches). B3 of the fourth table shows the same 131.947, and the fifth table shows about 21.97 (ft²) in B2. Just replace the input numbers with the size of your own cylinder.
In a formula, "PI()" is the Excel function that returns pi (3.14159…), "B1" and "B2" mean "use the number in that cell", "*" is multiplication, "/" is division and "^" is an exponent (how many times to multiply).
In the first table, for example, B3 shows about 56.549, B4 about 75.398 and B5 about 131.947 (in², since the inputs are in inches). B3 of the fourth table shows the same 131.947, and the fifth table shows about 21.97 (ft²) in B2. Just replace the input numbers with the size of your own cylinder.
How to calculate it in Google Sheets
Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.
Table to find the surface area of a cylinder (two bases + side)
| Radius of the base r (in) | 3 |
| Height h (in) | 4 |
| Both bases, 2πr² (in²) | =2*PI()*B1^2 |
| Lateral area, 2πrh (in²) | =2*PI()*B1*B2 |
| Surface area (in²) | =B3+B4 |
Table to find the base area (one circle)
| Radius of the base r (in) | 3 |
| Base area, πr² (in²) | =PI()*B1^2 |
Table to find the lateral area
| Radius of the base r (in) | 3 |
| Height h (in) | 4 |
| Lateral area, 2πrh (in²) | =2*PI()*B1*B2 |
Table to find it with the combined formula 2πr(r + h)
| Radius of the base r (in) | 3 |
| Height h (in) | 4 |
| Surface area, 2πr(r+h) (in²) | =2*PI()*B1*(B1+B2) |
Table to convert square inches to square feet
| Surface area in in² | 3163 |
| Surface area in ft² | =B1/144 |
The same formulas as in Excel (including the PI() function) work as is. Copy the whole table, paste it into cell A1, and replace the input numbers with the size of your own cylinder.
How to calculate it in Python
import math
radius = 3 # radius of the base (inches in this example)
height = 4 # height (same unit as the radius)
base_area = math.pi * radius ** 2 # base area (one circle, in2 in this example)
bases_area = base_area * 2 # both bases
lateral_area = 2 * math.pi * radius * height # lateral area (circumference x height)
surface_area = bases_area + lateral_area # surface area of the cylinder (= 2*pi*r*(r+h))
print(f"Base area (one circle): {base_area} in2")
print(f"Both bases: {bases_area} in2")
print(f"Lateral area: {lateral_area} in2")
print(f"Surface area: {surface_area} in2")
Runs with the standard library only. "math.pi" is pi, "**" is an exponent (squared here) and "*" is multiplication. Change the radius and height at the top and run it. This example uses inches; if you use feet, the surface area is in square feet, and multiplying it by 144 gives square inches.
How to write it in LaTeX and other math languages (copy and paste)
Surface area of a cylinder (the basic idea)
S = B × 2 + L
S = B \times 2 + L
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>S</mi>
<mo>=</mo>
<mi>B</mi>
<mo>×</mo>
<mn>2</mn>
<mo>+</mo>
<mi>L</mi>
</mrow>
</math>
S = B xx 2 + L
sBase*2 + sSide
S := sBase*2 + sSide;
S = s_base*2 + s_side;
S = B × 2 + L
Base area (the area of the base circle)
B = π × r²
B = \pi r^{2}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>B</mi>
<mo>=</mo>
<mi>π</mi>
<mo>⁢</mo>
<msup><mi>r</mi><mn>2</mn></msup>
</mrow>
</math>
B = pi r^2
Pi*r^2
sBase := Pi*r^2;
s_base = pi*r^2;
B = πr^2
Lateral area (the side unrolls into a rectangle)
L = 2 × π × r × h
L = 2\pi r h
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>L</mi>
<mo>=</mo>
<mn>2</mn>
<mo>⁢</mo>
<mi>π</mi>
<mo>⁢</mo>
<mi>r</mi>
<mo>⁢</mo>
<mi>h</mi>
</mrow>
</math>
L = 2 pi r h
2*Pi*r*h
sSide := 2*Pi*r*h;
s_side = 2*pi*r*h;
L = 2πrh
The combined formula (all in one step)
S = 2 × π × r × (r + h)
S = 2\pi r(r+h)
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>S</mi>
<mo>=</mo>
<mn>2</mn>
<mo>⁢</mo>
<mi>π</mi>
<mo>⁢</mo>
<mi>r</mi>
<mo>⁢</mo>
<mrow><mo>(</mo><mi>r</mi><mo>+</mo><mi>h</mi><mo>)</mo></mrow>
</mrow>
</math>
S = 2 pi r (r + h)
2*Pi*r*(r + h)
S := 2*Pi*r*(r + h);
S = 2*pi*r*(r + h);
S = 2πr(r + h)
Converting area units (in² to ft²)
S[ft²] = S[in²] ÷ 144
S_{\mathrm{ft^2}} = S_{\mathrm{in^2}} \div 144
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<msub><mi>S</mi><mrow><msup><mi mathvariant="normal">ft</mi><mn>2</mn></msup></mrow></msub>
<mo>=</mo>
<msub><mi>S</mi><mrow><msup><mi mathvariant="normal">in</mi><mn>2</mn></msup></mrow></msub>
<mo>÷</mo>
<mn>144</mn>
</mrow>
</math>
S_(ft^2) = S_(in^2) -: 144
sIn2/144
sFt2 := sIn2/144;
s_ft2 = s_in2/144;
S(ft²) = S(in²)/144
How to have ChatGPT do the calculation
You are a math calculation assistant. Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing). A cylinder has a base radius of 3 inches and a height of 4 inches. Find each of the following: 1. The base area (one circle) in square inches (in²) 2. The lateral area in square inches 3. The surface area of the cylinder (two bases + side) in square inches Show the formulas you used and the numbers from the execution result.
How to Use
-
1Enter your numbersType the numbers you want to calculate with into the input fields
-
2CalculatePress the "Calculate" button
-
3Check the resultThe result appears on the spot. The same page also explains the idea behind the calculation and the formula
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