Enter the radius and the height of the cylinder part of a capsule (a cylinder with a hemisphere on each end). You get the surface area (S = 4πr² + 2πrh), its breakdown and the area in cm² and m².
Enter the radius and the height in the same unit, as numbers only (no units. For example, for 3 cm enter "3"). The "height" is the height of the cylinder part in the middle only, without the hemisphere ends. If you only know the total length of the capsule, enter the total length minus the diameter (radius × 2).
Result and figure
Enter the radius and the height of the cylinder part 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?
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How to Use
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Formulas and figures
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Symbols and terms
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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)
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How to have ChatGPT do the calculation
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DataChef Features
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Related Features
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NumberChef Calculators List
What you can do on this page
- Enter the radius and the height of the cylinder part, and you get the surface area of a capsule (a cylinder with a hemisphere on each end) on the spot (\(S = 4\pi r^2 + 2\pi r h\))
- It also shows the breakdown: the surface area of the sphere part (\(4\pi r^2\)) and the area of the side of the cylinder (\(2\pi r h\))
- Area unit conversions are shown too: ft² when you enter inches, and in² when you enter feet. Switch "Units" to Metric to get m² and cm² instead
- The result is also shown as a 3D shape that you can turn with your mouse or by swiping
- 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). The "height" is the height of the cylinder part only, without the hemisphere ends (not the total length of the capsule). Other solids such as spheres and cylinders have their own pages.
What is this calculation used for?
Estimating the coating for medicine capsules (pharmaceuticals)
Medicine capsules are sometimes given a coating to hide the taste or smell, or so that they dissolve in the intestine instead of the stomach. The amount of coating needed is proportional to the surface area of the capsule.
A capsule with a radius of 0.14 in and a cylinder height of 0.47 in (about 0.75 in long in total, a common size) has a surface area of \(4\pi \times 0.14^2 + 2\pi \times 0.14 \times 0.47 \approx 0.66\,\mathrm{in^2}\). For 100,000 of them, the total area to coat is about 458 ft², which is the basis for how much coating material to prepare.
Estimating the paint for a gas storage tank (industry and equipment)
Pressure tanks for propane and other gases are often capsules, a cylinder with rounded ends, because that shape handles pressure well. When the rust-proof paint is redone, the amount of paint is estimated from the surface area.
A typical 500-gallon home propane tank has a radius of about 18.75 in (1.5625 ft) and a cylinder part about 82.5 in (6.875 ft) long. Its surface area is \(4\pi \times 1.5625^2 + 2\pi \times 1.5625 \times 6.875 \approx 98\,\mathrm{ft^2}\). Two coats come to about 196 ft², so with paint that covers about 350 ft² per gallon, one gallon is enough (in practice, add a little for the fittings and legs).
How much casing sausages need (food production)
A sausage is ground meat wrapped in a skin called a casing, and it is almost a capsule. The amount of casing depends on the surface area of each link.
A bratwurst-size link with a radius of 0.625 in and a cylinder part 4.25 in long (5.5 in long in total) has a surface area of \(4\pi \times 0.625^2 + 2\pi \times 0.625 \times 4.25 \approx 21.6\,\mathrm{in^2}\). Making 1,000 links takes about 150 ft² of casing (in practice, add a little extra for the tied ends).
Heat loss and insulation of a water heater tank (home and energy saving)
The tanks of tank-type water heaters are also cylinders with rounded ends, close to a capsule. Heat escapes from the surface of the tank, so both the heat loss and the amount of insulation blanket to wrap around it depend on the surface area.
A tank with a radius of 0.75 ft and a cylinder part 4 ft high has a surface area of \(4\pi \times 0.75^2 + 2\pi \times 0.75 \times 4 \approx 25.9\,\mathrm{ft^2}\). This formula also shows a basic rule of energy-saving design: for the same capacity, a shape with less surface area loses less heat.
Formulas and figures
Surface area of a capsule (the basic formula)
Figure
Standard notation
(the usual math form)
\(S\)
\(=\)
\(4\)
\(\times\)
\(\pi\)
\(\times\)
\(r\)
\(2\)
\(+\)
\(2\)
\(\times\)
\(\pi\)
\(\times\)
\(r\)
\(\times\)
\(h\)
In words
(symbols replaced with words)
⑨
\(S\): surface area of the capsule
\(=\)
①
\(4\): constant
\(\times\)
②
\(\pi\): pi
\(\times\)
③
\(r\): radius
④
squared (the number times itself)
\(+\)
⑤
\(2\): constant
\(\times\)
⑥
\(\pi\): pi
\(\times\)
⑦
\(r\): radius
\(\times\)
⑧
\(h\): height of the cylinder part
The formula in words
①
Multiply the
\(4\): constant
②
by
\(\pi\): pi
③
and by the
\(r\): radius
④
squared (the number times itself)
to get the surface area of the sphere part.
⑤
Multiply the
\(2\): constant
⑥
by
\(\pi\): pi
⑦
by the
\(r\): radius
⑧
and by the
\(h\): height of the cylinder part
to get the area of the side of the cylinder.
⑨
Add these two and you get the
\(S\): surface area of the capsule
Quick example
For a capsule with a radius of 3 in and a cylinder height of 10 in (16 in long in total), the surface area is
\(S\): surface area of the capsule
\(=\)
\(4\): constant
\(\times\)
\(\pi\): pi
\(\times\)
radius (3 in)
squared
\(+\)
\(2\): constant
\(\times\)
\(\pi\): pi
\(\times\)
radius (3 in)
\(\times\)
cylinder height (10 in)
\(4 \times \pi \times 3^2 = 36\pi\)
\(2 \times \pi \times 3 \times 10 = 60\pi\)
\(36\pi + 60\pi = 96\pi \approx 301.6\,\mathrm{in^2}\)
Key idea
A capsule is made of a cylinder in the middle and two hemispheres on the ends. The two hemispheres together make exactly one sphere, so the first half, \(4\pi r^2\), is the surface area of a sphere. The second half, \(2\pi r h\), is the area of the side of the cylinder. The top and bottom circles of the cylinder are joined to the hemispheres on the inside, so they are not part of the surface area.
The "height \(h\)" is the height of the cylinder part only. The total length of the capsule is \(h + 2r\), so if you know the total length, find \(h\) as total length − diameter before using the formula.
Surface area of the sphere part (the two hemisphere ends make one sphere)
Figure
Standard notation
(the usual math form)
\(S_1\)
\(=\)
\(4\)
\(\times\)
\(\pi\)
\(\times\)
\(r\)
\(2\)
In words
(symbols replaced with words)
⑤
\(S_1\): surface area of the sphere part
\(=\)
①
\(4\): constant
\(\times\)
②
\(\pi\): pi
\(\times\)
③
\(r\): radius
④
squared (the number times itself)
The formula in words
①
Multiply the
\(4\): constant
②
by
\(\pi\): pi
③
and by the
\(r\): radius
④
squared (the number times itself)
⑤
and you get the
\(S_1\): surface area of the sphere part
Quick example
For a capsule with a radius of 3 in, the surface area of the sphere part is
\(S_1\): surface area of the sphere part
\(=\)
\(4\): constant
\(\times\)
\(\pi\): pi
\(\times\)
radius (3 in)
squared
\(3 \times 3 = 9\)
\(4 \times \pi \times 9 = 36\pi \approx 113.1\,\mathrm{in^2}\)
Key idea
Each end of a capsule is a hemisphere, a sphere cut exactly in half. The left and right hemispheres together make one whole sphere again, so you can use the surface area of a sphere, \(S = 4\pi r^2\), just as it is (half the area, twice, is one whole sphere).
The form "π × radius squared, times 4" says that the surface area of a sphere is exactly 4 times the area of a circle with the same radius.
Area of the side of the cylinder (it unrolls into a rectangle)
Figure
Standard notation
(the usual math form)
\(S_2\)
\(=\)
\(2\)
\(\times\)
\(\pi\)
\(\times\)
\(r\)
\(\times\)
\(h\)
In words
(symbols replaced with words)
⑤
\(S_2\): area of the side of the cylinder
\(=\)
①
\(2\): constant
\(\times\)
②
\(\pi\): pi
\(\times\)
③
\(r\): radius
\(\times\)
④
\(h\): height of the cylinder part
The formula in words
①
Multiply the
\(2\): constant
②
by
\(\pi\): pi
③
and by the
\(r\): radius
(this is the circumference)
④
then multiply by the
\(h\): height of the cylinder part
⑤
and you get the
\(S_2\): area of the side of the cylinder
Quick example
For a capsule with a radius of 3 in and a cylinder height of 10 in, the area of the side of the cylinder is
\(S_2\): area of the side of the cylinder
\(=\)
\(2\): constant
\(\times\)
\(\pi\): pi
\(\times\)
radius (3 in)
\(\times\)
cylinder height (10 in)
\(2 \times \pi \times 3 = 6\pi\)
\(6\pi \times 10 = 60\pi \approx 188.5\,\mathrm{in^2}\)
Key idea
Cut the side of a cylinder straight down with scissors and unroll it, and it becomes one rectangle (its net). The width is the circumference of the base (\(2\pi r\)) and the height is \(h\), so the area of the side is "circumference × height" = \(2\pi r h\).
For the surface area of an ordinary cylinder, you would add the top and bottom circles (\(2\pi r^2\) for the two). In a capsule, those circles are joined to the hemisphere ends and are not on the outside, so you count only the side.
Converting area units (in² → 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
\(S_{\mathrm{ft^2}}\): area in ft²
\(=\)
①
\(S_{\mathrm{in^2}}\): area in in²
\(\div\)
②
\(144\): square inches in 1 ft²
The formula in words
①
Take the
\(S_{\mathrm{in^2}}\): area in in²
②
divide it by
\(144\): square inches in 1 ft²
③
and you get the
\(S_{\mathrm{ft^2}}\): area in ft²
Quick example
The surface area of the capsule with a radius of 3 in and a cylinder height of 10 in, about 301.6 in², in ft² is
area in ft²
\(=\)
area in in² (301.6)
\(\div\)
square inches in 1 ft² (144)
\(301.6 \div 144 \approx 2.094\,\mathrm{ft^2}\)
Key idea
The conversion factor for area is the square of the factor for length. Since \(1\,\mathrm{ft} = 12\,\mathrm{in}\), \(1\,\mathrm{ft^2}\) is \(12 \times 12 = 144\,\mathrm{in^2}\). It is 144 times, not 12 times, which is easy to get wrong.
To go back from ft² to in², multiply by 144. This calculator shows both the ft² value when you enter inches and the in² value when you enter feet.
The surface area of a capsule is "sphere part (4πr²) + side of the cylinder (2πrh)". The key points are that the two hemisphere ends make exactly one sphere, and that only the side of the cylinder is counted. Use the height of the cylinder part only for h, not the total length of the capsule.
Symbols and terms
Symbols
| \(S\) | ess | A common symbol for area, from "surface". On this page it is the surface area of the whole capsule (some textbooks write SA). |
| \(S_1\) | S sub one | The symbol on this page for the surface area of the sphere part (\(4\pi r^2\)). The small 1 at the lower right is a subscript for "part 1". |
| \(S_2\) | S sub two | The symbol on this page for the area of the side of the cylinder (\(2\pi r h\)). The small 2 at the lower right is a subscript for "part 2". |
| \(r\) | ar | The radius of the capsule (half its thickness), shared by the cylinder part and the hemisphere ends. It comes from the first letter of "radius" and is half the diameter. |
| \(h\) | aitch | The height of the cylinder part (the length of the straight part in the middle, without the hemisphere ends), from the first letter of "height". The total length of the capsule is \(h + 2r\). |
| \(\pi\) | pi | The Greek letter for the circumference ÷ diameter of a circle. It is \(3.14159\ldots\) and never ends. The 3.14 used in simple calculations is an approximation of it. |
| \(r^2\) | r squared | The number \(r\) multiplied by itself (\(r \times r\)). The small 2 at the upper right is an exponent that says "use it as a factor twice". |
| \(\mathrm{in^2}\) | square inch | A unit of area. A square 1 in on each side has an area of 1 in². |
| \(\mathrm{ft^2}\) | square foot | A unit of area. A square 1 ft on each side has an area of 1 ft². \(1\,\mathrm{ft^2} = 144\,\mathrm{in^2}\). |
Terms
| capsule | A solid made by attaching a hemisphere of the same radius to each end of a cylinder. It is a common shape in everyday life and industry, from medicine capsules to gas storage tanks. |
| hemisphere | A sphere cut exactly in half by a plane through its center. Each end of a capsule is a hemisphere, and two of them together make one sphere. |
| surface area | The area of the whole outside of a solid, like the area of paper needed to wrap it. For a capsule, it is the sphere part plus the side of the cylinder. |
| lateral area | The area of the side all around a prism-like solid (also called the lateral surface area). For a cylinder, the side unrolls into a rectangle, and its area is circumference × height (\(2\pi r h\)). |
| net | The flat pattern you get by cutting a solid open and laying it flat. The net of the side of a cylinder is a rectangle with width = circumference and height = height, which is where the lateral area formula comes from. |
| circumference | The distance around a circle. It is diameter × π (\(2\pi r\)). In the area of the side of a cylinder, it is the width of the rectangle. |
| pi | The number that tells how many times the circumference of a circle is its diameter. It is the same for a circle of any size and is written with the symbol \(\pi\) (about 3.14). |
| unit conversion | Writing the same amount in a different unit. For area, the key is that the conversion factor is the square of the length factor (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) |
|
| Units of area (Grades 3–6) |
|
| Nets and lateral area of a cylinder (Grades 6–8) |
|
| Surface area of a sphere (high school geometry) |
|
| Pi and expressions with letters (Grades 6–7) |
|
| Working with decimals (Grades 5–6) |
|
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 capsule
| Radius r (in) | 3 |
| Cylinder height h (in) | 10 |
| Capsule surface area (in²) | =4*PI()*B1^2+2*PI()*B1*B2 |
Table to find the surface area of the sphere part
| Radius r (in) | 3 |
| Sphere part, 4πr² (in²) | =4*PI()*B1^2 |
Table to find the area of the side of the cylinder
| Radius r (in) | 3 |
| Cylinder height h (in) | 10 |
| Side of the cylinder, 2πrh (in²) | =2*PI()*B1*B2 |
Table to convert in² to ft²
| Area in in² | 301.6 |
| 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 formula in the last row calculates from them automatically.
In a formula, "PI()" is the Excel function that returns π (3.14159…), "B1" and "B2" tell the formula to use the number in that cell, "*" is multiplication, "/" is division and "^" raises to a power (how many times to multiply).
In the first table, for example, B3 shows about 301.593 (in in², because the inputs are in inches). The second table shows about 113.097 in B2, the third about 188.496 in B3, and the fourth about 2.094 in B2. Just replace the input numbers with the size of your own capsule.
In a formula, "PI()" is the Excel function that returns π (3.14159…), "B1" and "B2" tell the formula to use the number in that cell, "*" is multiplication, "/" is division and "^" raises to a power (how many times to multiply).
In the first table, for example, B3 shows about 301.593 (in in², because the inputs are in inches). The second table shows about 113.097 in B2, the third about 188.496 in B3, and the fourth about 2.094 in B2. Just replace the input numbers with the size of your own capsule.
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 capsule
| Radius r (in) | 3 |
| Cylinder height h (in) | 10 |
| Capsule surface area (in²) | =4*PI()*B1^2+2*PI()*B1*B2 |
Table to find the surface area of the sphere part
| Radius r (in) | 3 |
| Sphere part, 4πr² (in²) | =4*PI()*B1^2 |
Table to find the area of the side of the cylinder
| Radius r (in) | 3 |
| Cylinder height h (in) | 10 |
| Side of the cylinder, 2πrh (in²) | =2*PI()*B1*B2 |
Table to convert in² to ft²
| Area in in² | 301.6 |
| 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 capsule.
How to calculate it in Python
import math
radius = 3 # radius (inches in this example)
height = 10 # height of the cylinder part (same unit as the radius; not the total length)
sphere_area = 4 * math.pi * radius ** 2 # surface area of the sphere part (two hemispheres = one sphere; in2 in this example)
lateral_area = 2 * math.pi * radius * height # area of the side of the cylinder (in2 in this example)
surface_area = sphere_area + lateral_area # surface area of the capsule (in2 in this example)
print(f"Sphere part: {sphere_area} in2")
print(f"Side of the cylinder: {lateral_area} in2")
print(f"Capsule surface area: {surface_area} in2")
Runs with the standard library only. "math.pi" is π, "**" raises to a power (squared) and "*" is multiplication. Change the radius and the cylinder height at the top and run it. (This example uses inches. Divide the result by 144 to get ft². If you enter feet, the area is in ft².)
How to write it in LaTeX and other math languages (copy and paste)
Surface area of a capsule (the basic formula)
S = 4πr² + 2πrh
S = 4 \pi r^{2} + 2 \pi r h
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>S</mi>
<mo>=</mo>
<mn>4</mn>
<mo>⁢</mo>
<mi>π</mi>
<mo>⁢</mo>
<msup><mi>r</mi><mn>2</mn></msup>
<mo>+</mo>
<mn>2</mn>
<mo>⁢</mo>
<mi>π</mi>
<mo>⁢</mo>
<mi>r</mi>
<mo>⁢</mo>
<mi>h</mi>
</mrow>
</math>
S = 4 pi r^2 + 2 pi r h
4*Pi*r^2 + 2*Pi*r*h
S := 4*Pi*r^2 + 2*Pi*r*h;
S = 4*pi*r^2 + 2*pi*r*h;
S = 4πr^2 + 2πrh
Surface area of the sphere part (the two hemisphere ends make one sphere)
S₁ = 4πr²
S_1 = 4 \pi r^{2}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<msub><mi>S</mi><mn>1</mn></msub>
<mo>=</mo>
<mn>4</mn>
<mo>⁢</mo>
<mi>π</mi>
<mo>⁢</mo>
<msup><mi>r</mi><mn>2</mn></msup>
</mrow>
</math>
S_1 = 4 pi r^2
4*Pi*r^2
S1 := 4*Pi*r^2;
s1 = 4*pi*r^2;
S_1 = 4πr^2
Area of the side of the cylinder (it unrolls into a rectangle)
S₂ = 2πrh
S_2 = 2 \pi r h
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<msub><mi>S</mi><mn>2</mn></msub>
<mo>=</mo>
<mn>2</mn>
<mo>⁢</mo>
<mi>π</mi>
<mo>⁢</mo>
<mi>r</mi>
<mo>⁢</mo>
<mi>h</mi>
</mrow>
</math>
S_2 = 2 pi r h
2*Pi*r*h
S2 := 2*Pi*r*h;
s2 = 2*pi*r*h;
S_2 = 2πrh
Converting area units (in² → 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 capsule (a cylinder with a hemisphere on each end) has a radius of 3 in and a cylinder part 10 in high. Find each of the following: 1. The surface area of the sphere part (4πr²) in in² 2. The area of the side of the cylinder (2πrh) in in² 3. The surface area of the whole capsule in in² 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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Related Features
Sphere Volume Calculator (V = 4/3 × π × r³)
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Cylinder Volume Calculator (V = πr²h)
Calculates the volume of a cylinder, the middle part of a capsule, and explains the idea of base area × height
Circle Calculator (Radius, Diameter, Circumference and Area)
Explains in detail the circumference (2πr) behind the side area of a cylinder. Also handy for getting the radius from the diameter
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