Enter the two values you know out of the base radius r (the radius of the cut circle), the sphere radius R and the height h. The third value and the volume are calculated.
Enter lengths as numbers of 0 or more, all in the same unit (numbers only, no units). Leave the remaining value blank. When you start from the base radius and the sphere radius, two answers may be shown.
Result and figure
Enter two of the three values in the fields on the left and press "Calculate". The result will appear here.
Table of Contents
-
What you can do on this page
-
What is this calculation used for?
-
How to Use
-
Formulas and figures
-
Spherical cap volume formula (from the sphere radius and the height)
-
Formula for the sphere radius \(R\) (from the base radius and the height)
-
Formula for the height \(h\) (from the base radius and the sphere radius, two answers)
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Formula for the base radius \(r\) (from the sphere radius and the height)
-
-
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)
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How to have ChatGPT do the calculation
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DataChef Features
-
Related Features
-
NumberChef Calculators List
What you can do on this page
- Find the volume of a spherical cap on the spot. A spherical cap is the piece you get when you slice a sphere with one flat cut. It looks like a dome or an upside-down bowl
- Of the base radius \(r\) (the radius of the cut circle), the sphere radius \(R\) and the height \(h\), enter the two you know, and the third is calculated automatically
- When you start from the base radius and the sphere radius, there can be two answers, depending on whether the cut is above or below the center of the sphere. Both are shown
- The resulting cap is also drawn in 3D (drag to rotate it). The whole sphere before the cut is shown with thin lines
- A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page
Enter all three values in the same unit (all in inches, for example). The volume comes out in that unit cubed (in³ if you enter inches, ft³ if you enter feet). To see liters and m³ in the result instead, switch "Units" above the calculator to Metric.
What is this calculation used for?
Finding how much liquid is left in a spherical tank from its depth (industry and infrastructure)
In a spherical tank for oil or liquefied gas, the liquid inside takes the shape of a spherical cap. So if you measure the liquid depth \(h\), this formula tells you how much is left. For example, if a spherical tank with a radius of 6 ft holds liquid 2 ft deep, the amount is \(\pi \times 2^2 \times (3 \times 6 - 2) \div 3 \approx 67.0\,\mathrm{ft^3}\) (about 501 gallons).
Turning a depth reading from a sensor into a volume is the most typical real-world use of this formula.
Estimating how much is in a round bowl (cooking)
Pour soup 1.5 in deep into a nearly hemispherical bowl with a radius of 3 in. The soup takes the shape of a spherical cap, so the amount is \(\pi \times 1.5^2 \times (3 \times 3 - 1.5) \div 3 = 5.625\pi \approx 17.7\,\mathrm{in^3}\) (about 9.8 fl oz).
A full bowl (a hemisphere) holds about 31.3 fl oz, so the math shows that "half the depth is much less than half the amount" (less than a third). It is the most familiar example, and it helps with serving sizes and recipe amounts.
Estimating the space inside a dome roof or a planetarium (architecture)
A dome roof or ceiling can be estimated as a spherical cap. For a dome 400 ft across (a cut circle with a radius of 200 ft) and 100 ft tall, the original sphere has a radius of \(R = (100^2 + 200^2) \div (2 \times 100) = 250\,\mathrm{ft}\), and the space inside is about 6.8 million ft³.
This volume is the starting point for designing heating and cooling, ventilation and acoustics (real domes are not always exact spherical caps, so it is only an estimate).
Measuring the volume of a droplet on a surface (printing, coating and semiconductor technology)
Small droplets sitting on a surface, like a water drop on a leaf or an ink drop from an inkjet printer, take almost the shape of a spherical cap because of surface tension. In research and manufacturing, people photograph the droplet from the side, measure the radius of its base and its height, and find its volume with this formula.
It is also used to measure how easily a material gets wet (the contact angle), so the formula is useful even at the microscopic scale.
Working with the curved part of a lens (optics)
The bulging face of an eyeglass or camera lens is part of a sphere, and the bulging part itself is a spherical cap. This formula has long been used to estimate how much material to grind away when making a lens, and how much material a lens needs.
For thin dome shapes like contact lenses, the relation between the base radius, the sphere radius and the height (Formulas 2 to 4 on this page) is the basis of the design.
Formulas and figures
Spherical cap volume formula (from the sphere radius and the height)
Figure
Standard notation
(the usual math form)
\(V\)
\(=\)
\(\pi\)
\(\times\)
\(h\)
\(2\)
\(\times\)
\((\)
\(3 \times\)
\(R\)
\(-\)
\(h\)
\()\)
\(\div 3\)
In words
(symbols replaced with words)
⑥
\(V\): cap volume
\(=\)
③
\(\pi\): pi
\(\times\)
①
\(h\): height
②
squared (the number times itself)
\(\times\)
\((\)
\(3 \times\)
④
\(R\): sphere radius
\(-\)
\(h\): height
\()\)
⑤
\(\div 3\)
The formula in words
①
Take the
\(h\): height
,
②
find its
square (the number times itself)
,
③
multiply by
\(\pi\): pi (about 3.14)
,
④
then multiply by "\(3R - h\)", which is 3 times the
\(R\): sphere radius
minus the height \(h\),
⑤
and finally divide by 3 to get the
⑥
\(V\): cap volume
Quick example
Cut a sphere with a radius of 2 in so that the cap is 1 in tall (like an upside-down bowl). The volume of this cap is
\(V\): cap volume
\(=\)
\(\pi\): pi
\(\times\)
height (1 in)
squared
\(\times\)
\((\)
\(3 \times\)
sphere radius (2 in)
\(-\)
height (1 in)
\()\)
\(\div 3\)
\(\pi \times 1^{2} \times (3 \times 2 - 1) \div 3 = \pi \times 5 \div 3 = \dfrac{5}{3}\pi\)
\(\dfrac{5}{3}\pi \approx \dfrac{5}{3} \times 3.14 \approx 5.2\,\mathrm{in^3}\)
Key idea
The most common mistake is to double the height \(h\) instead of squaring it.
This formula uses only the height \(h\) and the sphere radius \(R\). The base radius \(r\) does not appear, because once \(h\) and \(R\) are fixed, \(r\) is fixed too. For a hemisphere (\(h = R\)) you get \(V = \dfrac{2}{3}\pi R^3\), exactly half the volume of the sphere. With \(h = 2R\) (the full diameter) you get \(\dfrac{4}{3}\pi R^3\), the volume of the whole sphere.
If you want the volume straight from the base radius \(r\) and the height \(h\), the same formula can be rewritten as \(V = \pi h (3r^2 + h^2) \div 6\).
Formula for the sphere radius \(R\) (from the base radius and the height)
Figure
Standard notation
(the usual math form)
\(R\)
\(=\)
\((\)
\(h\)
\(2\)
\(+\)
\(r\)
\(2\)
\()\)
\(\div\)
\((\)
\(2 \times\)
\(h\)
\()\)
In words
(symbols replaced with words)
⑤
\(R\): sphere radius
\(=\)
\((\)
①
\(h\): height
②
squared (the number times itself)
\(+\)
③
\(r\): base radius
squared
\()\)
\(\div\)
\((\)
\(2 \times\)
④
\(h\): height
\()\)
The formula in words
①
Take the
\(h\): height
,
②
find its
square (the number times itself)
,
③
square the
\(r\): base radius
too, and add the two,
④
then divide the sum by "\(2h\)", which is 2 times the
\(h\): height
, to get the
⑤
\(R\): sphere radius
Quick example
For a dome-shaped object whose cut circle has a radius of 1 ft and whose height is 0.5 ft, the radius of the original sphere is
\(R\): sphere radius
\(=\)
\((\)
height (0.5 ft)
squared
\(+\)
base radius (1 ft)
squared
\()\)
\(\div\)
\((\)
\(2 \times\)
height (0.5 ft)
\()\)
\(R = (0.5^{2} + 1^{2}) \div (2 \times 0.5) = 1.25 \div 1\)
\(1.25 \div 1 = 1.25\,\mathrm{ft}\)
Key idea
This formula is really the Pythagorean theorem. Connect three points: the center of the sphere, the center of the cut circle and a point on the edge of the cut. They form a right triangle, so \(r^2 + (R - h)^2 = R^2\). Solving this for \(R\) gives the formula.
What makes this formula interesting is that you can find the size of the whole sphere from just a piece of it. It is the same idea as estimating the size of a broken bowl from one of its pieces.
Formula for the height \(h\) (from the base radius and the sphere radius, two answers)
Figure
Standard notation
(the usual math form)
\(h\)
\(=\)
\(R\)
\(\pm\)
\(\sqrt{R^{2} - r^{2}}\)
In words
(symbols replaced with words)
④
\(h\): cap height
\(=\)
①
\(R\): sphere radius
③
\(\pm\)
②
square root of the difference of squares \(R^2 - r^2\)
The formula in words
①
Start from the
\(R\): sphere radius
,
②
find the
square root of the difference of squares \(R^2 - r^2\)
(square the sphere radius \(R\), subtract the square of the base radius \(r\), and take the square root),
③
then either subtract it or add it (the sign ± stands for "both − and + are possible"). This gives the
④
\(h\): cap height
(usually two answers)
Quick example
When the cut circle has a radius of 1 ft and the original sphere has a radius of 2 ft, the height of the cap is
\(h\): cap height
\(=\)
sphere radius (2 ft)
\(\pm\)
square root of "2² − 1²"
\(\sqrt{2^{2} - 1^{2}} = \sqrt{4 - 1} = \sqrt{3} \approx 1.732\)
\(h = 2 - 1.732 = 0.268\,\mathrm{ft}\)
\(h = 2 + 1.732 = 3.732\,\mathrm{ft}\)
Key idea
There are two answers because the same size of cut circle can make two different caps: one cut above the center of the sphere (a shallow bowl) and one cut below the center (a deep jar).
Only when \(r = R\) (the cut passes through the center of the sphere) is \(\sqrt{R^2 - r^2} = 0\), and then there is just one answer: a hemisphere with \(h = R\). Also, \(r > R\) is not possible, because a cut circle cannot be larger than the sphere.
Formula for the base radius \(r\) (from the sphere radius and the height)
Figure
Standard notation
(the usual math form)
\(r\)
\(=\)
\(\sqrt{2Rh - h^{2}}\)
In words
(symbols replaced with words)
②
\(r\): base radius
\(=\)
①
square root of \(2Rh - h^2\)
The formula in words
①
Multiply 2 times the sphere radius \(R\) by the height \(h\), subtract the square of the height \(h\), and take the
square root of \(2Rh - h^2\)
to get the
②
\(r\): base radius
Quick example
For a cap cut from a sphere with a radius of 2 in, with a height of 1 in, the radius of the cut circle is
\(r\): base radius
\(=\)
square root of "2 × 2 × 1 − 1²"
\(r = \sqrt{2 \times 2 \times 1 - 1^{2}} = \sqrt{3} \approx 1.73\,\mathrm{in}\)
Key idea
This one also comes from the Pythagorean theorem. It is \(r^2 = R^2 - (R - h)^2\) with the right side expanded and simplified. You can also factor it and write \(r = \sqrt{h(2R - h)}\).
When the height is the full diameter (\(h = 2R\)), you get \(r = 0\): there is no cut, and you are back to the whole sphere. In the example of Formula 1 (radius 2 in, height 1 in), \(r = \sqrt{3} \approx 1.73\,\mathrm{in}\). Put this \(r = \sqrt{3}\) and \(R = 2\) into Formula 3 and you get \(h = 2 \pm \sqrt{4 - 3} = 1\) or \(3\). One of the two answers is the original height of 1 in, just as it should be.
The volume of a spherical cap is "V = π × height squared × (3 × sphere radius − height) ÷ 3". It depends only on the height and the sphere radius. The base radius, the sphere radius and the height are linked by the Pythagorean theorem, so if you know any two of them, you can calculate the third.
Symbols and terms
Symbols
| \(V\) | vee | A common symbol for volume, from the first letter of "volume". On this page it stands for the volume of the spherical cap. |
| \(r\) | lowercase r | The base radius, that is, the radius of the circle you get when you cut the sphere with a plane. From the first letter of "radius". |
| \(R\) | capital R | The radius of the original sphere, the whole ball before the cut. It is written as a capital letter to tell it apart from the radius of the cut circle, lowercase \(r\). |
| \(h\) | aitch | The height of the cap, the distance from the flat cut to the highest point of the cap. From the first letter of "height". |
| \(\pi\) | pi | The ratio of a circle's circumference to its diameter. It is about 3.14159… and its digits never end. In school, 3.14 is often used. |
| \(\pm\) | plus or minus | A symbol for "there are two cases, one with + and one with −". On this page it shows that the height can have two answers. |
| \(\sqrt{\phantom{a}}\) | square root (radical sign) | The symbol for a square root (a number that gives the original number when squared). Example: \(\sqrt{9} = 3\), because \(3^2 = 9\). |
| \(\mathrm{in^3}\) | cubic inches | A unit of volume. A cube with 1-inch sides has a volume of 1 in³. Do not mix it up with in² (square inches), the unit of area. |
| \(\mathrm{gal}\) | gallons | A unit often used for amounts of liquid such as water. One US gallon is \(1\,\mathrm{gal} = 231\,\mathrm{in^3}\) (a gallon jug of milk holds exactly 1 gal). |
Terms
| spherical cap | The solid piece you get when you slice a sphere with one flat cut. It looks like an upside-down bowl. This calculator finds the volume of this solid. |
| curved surface | The rounded part of the surface of a spherical cap. In English, "spherical cap" can refer to either the solid or just this curved surface; on this page it refers to the solid. |
| hemisphere | The solid you get by cutting a sphere with a plane through its center. It is a special spherical cap with height = sphere radius (\(h = R\)), and its volume is exactly half of the whole sphere. |
| base | The flat face of the spherical cap, that is, the cut circle. The radius of this circle is \(r\). |
| Pythagorean theorem | The theorem that in a right triangle, "hypotenuse squared = the sum of the squares of the other two sides". For a spherical cap, \(r^2 + (R - h)^2 = R^2\), and every formula that finds one of \(r\), \(R\) and \(h\) from the other two comes from this. |
| square root | A number that gives the original number when squared. The square roots of 9 are 3 and −3, but for lengths only the positive one is used (\(\sqrt{9} = 3\)). |
| volume | The amount of space a solid takes up, given as a number. It is measured by how many cubes with 1-inch sides (1 in³) would fill it. |
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.
| Circles and pi (Grade 7) |
|
| What volume is and its units (Grade 5) |
|
| Exponents (Grade 6) |
|
| Volume of a sphere (Grade 8) |
|
| Square roots (Grade 8) |
|
| The Pythagorean theorem (Grade 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 volume from the sphere radius and the height
| Sphere radius R | 2 |
| Height h | 1 |
| Cap volume | =PI()*B2^2*(3*B1-B2)/3 |
Table to find the sphere radius from the base radius and the height
| Base radius r | 1 |
| Height h | 0.5 |
| Sphere radius R | =(B2^2+B1^2)/(2*B2) |
Table to find the height (two answers) from the base radius and the sphere radius
| Base radius r | 1 |
| Sphere radius R | 2 |
| Height (solution 1) | =B2-SQRT(B2^2-B1^2) |
| Height (solution 2) | =B2+SQRT(B2^2-B1^2) |
Table to find the base radius from the sphere radius and the height
| Sphere radius R | 2 |
| Height h | 1 |
| Base radius r | =SQRT(2*B1*B2-B2^2) |
After pasting, column A holds the item names and column B holds the numbers. The upper rows are your inputs, and the formula in the last row calculates automatically from them.
"PI()" is the function for pi (π) and "SQRT()" is the function for a square root. "^" is a power (squared), "*" is multiplication and "/" is division.
In the first table, for example, B3 shows about 5.24 (in³ for a radius of 2 in and a height of 1 in). The third table is the case with two answers: B3 shows about 0.268 and B4 about 3.732. Just replace the input numbers with your own.
"PI()" is the function for pi (π) and "SQRT()" is the function for a square root. "^" is a power (squared), "*" is multiplication and "/" is division.
In the first table, for example, B3 shows about 5.24 (in³ for a radius of 2 in and a height of 1 in). The third table is the case with two answers: B3 shows about 0.268 and B4 about 3.732. Just replace the input numbers with your own.
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 volume from the sphere radius and the height
| Sphere radius R | 2 |
| Height h | 1 |
| Cap volume | =PI()*B2^2*(3*B1-B2)/3 |
Table to find the sphere radius from the base radius and the height
| Base radius r | 1 |
| Height h | 0.5 |
| Sphere radius R | =(B2^2+B1^2)/(2*B2) |
Table to find the height (two answers) from the base radius and the sphere radius
| Base radius r | 1 |
| Sphere radius R | 2 |
| Height (solution 1) | =B2-SQRT(B2^2-B1^2) |
| Height (solution 2) | =B2+SQRT(B2^2-B1^2) |
Table to find the base radius from the sphere radius and the height
| Sphere radius R | 2 |
| Height h | 1 |
| Base radius r | =SQRT(2*B1*B2-B2^2) |
The same formulas as in Excel (including the PI() and SQRT() functions) work as is. Copy the whole table, paste it into cell A1, and replace the input numbers with your own.
How to calculate it in Python
import math
ball_radius = 2 # sphere radius R (in inches in this example)
height = 1 # cap height h
volume = math.pi * height ** 2 * (3 * ball_radius - height) / 3 # cap volume (input unit cubed)
base_radius = math.sqrt(2 * ball_radius * height - height ** 2) # base radius r (radius of the cut circle)
print(f"Cap volume: {volume} in3")
print(f"Base radius: {base_radius} in")
Runs with the standard library only. "math.pi" is pi (π), "math.sqrt" is the square root, "**" is a power (squared), "*" is multiplication and "/" is division. Change the sphere radius and the height at the top and run it (this example uses inches).
How to write it in LaTeX and other math languages (copy and paste)
Spherical cap volume formula (from the sphere radius and the height)
V = πh²(3R − h)/3
V = \dfrac{1}{3}\pi h^{2}(3R - h)
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>V</mi>
<mo>=</mo>
<mfrac><mn>1</mn><mn>3</mn></mfrac>
<mi>π</mi>
<msup><mi>h</mi><mn>2</mn></msup>
<mo>(</mo>
<mn>3</mn><mi>R</mi>
<mo>−</mo>
<mi>h</mi>
<mo>)</mo>
</mrow>
</math>
V = (1/3) pi h^2 (3R - h)
Pi*h^2*(3*R - h)/3
V := Pi*h^2*(3*R - h)/3;
V = pi*h^2*(3*R - h)/3;
V = (1/3)πh^2(3R - h)
Formula for the sphere radius \(R\) (from the base radius and the height)
R = (h² + r²)/(2h)
R = \dfrac{h^{2} + r^{2}}{2h}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>R</mi>
<mo>=</mo>
<mfrac>
<mrow><msup><mi>h</mi><mn>2</mn></msup><mo>+</mo><msup><mi>r</mi><mn>2</mn></msup></mrow>
<mrow><mn>2</mn><mi>h</mi></mrow>
</mfrac>
</mrow>
</math>
R = (h^2 + r^2)/(2h)
(h^2 + r^2)/(2*h)
R := (h^2 + r^2)/(2*h);
R = (h^2 + r^2)/(2*h);
R = (h^2 + r^2)/(2h)
Formula for the height \(h\) (from the base radius and the sphere radius, two answers)
h = R ± √(R² − r²)
h = R \pm \sqrt{R^{2} - r^{2}}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>h</mi>
<mo>=</mo>
<mi>R</mi>
<mo>±</mo>
<msqrt>
<mrow>
<msup><mi>R</mi><mn>2</mn></msup>
<mo>−</mo>
<msup><mi>r</mi><mn>2</mn></msup>
</mrow>
</msqrt>
</mrow>
</math>
h = R +- sqrt(R^2 - r^2)
Solve[r^2 + (R - h)^2 == R^2, h]
solve(r^2 + (R - h)^2 = R^2, h);
h = [R - sqrt(R^2 - r^2), R + sqrt(R^2 - r^2)];
h = R ± √(R^2 - r^2)
Formula for the base radius \(r\) (from the sphere radius and the height)
r = √(2Rh − h²)
r = \sqrt{2Rh - h^{2}}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>r</mi>
<mo>=</mo>
<msqrt>
<mrow>
<mn>2</mn><mi>R</mi><mi>h</mi>
<mo>−</mo>
<msup><mi>h</mi><mn>2</mn></msup>
</mrow>
</msqrt>
</mrow>
</math>
r = sqrt(2Rh - h^2)
Sqrt[2*R*h - h^2]
r := sqrt(2*R*h - h^2);
r = sqrt(2*R*h - h^2);
r = √(2Rh - h^2)
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 sphere with a radius of 2 in is cut by a plane so that the piece (a spherical cap) is 1 in tall, measured from the top. Find each of the following: 1. The volume of this spherical cap in in³ (V = π × h² × (3R − h) ÷ 3) 2. The radius of the cut circle (the base) in inches (r = √(2Rh − h²)) 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)
Finds the volume of the whole sphere, before any part is cut off
Pythagorean Theorem Calculator (Hypotenuse and Legs of a Right Triangle)
Works with the right-triangle theorem behind the relation between r, R and h of a spherical cap
Nth Root Calculator (Square Root, Cube Root, nth Root)
Finds square roots (√) like the ones that appear when you solve for the height or the base radius
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