Enter the radius of the sphere. The volume (4/3 × π × radius cubed) is shown, along with conversions to liters (L) and cubic meters (m³).
Enter only a number of 0 or more (no units. For a radius of 12 cm, enter "12"). If you only know the diameter, divide it by 2 and enter that (the radius).
Result and figure
Enter the radius of the sphere in the field 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 of a sphere, and its volume (volume = \(\dfrac{4}{3} \times \pi \times\) radius cubed) is calculated on the spot
- The sphere is also drawn as a 3D shape (drag it to turn it around)
- The result also shows the volume in US gallons and cubic feet when you enter inches. Switch "Units" to Metric to get liters and cubic meters instead
- Also explains what to do when you only know the diameter (radius = diameter ÷ 2, or \(V = \pi d^3 \div 6\))
- 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 in one unit (for example, inches). The volume comes out in that unit cubed (in³ if you enter inches, ft³ if you enter feet). The volumes of other solids, such as cylinders and cones, have their own pages.
What is this calculation used for?
Comparing balls by volume (sports)
A size 5 soccer ball is about 8.7 in across, so with a radius of 4.35 in its volume is \(\dfrac{4}{3}\pi \times 4.35^3 \approx 345\,\mathrm{in^3}\), about 1.5 gallons. A baseball (about 2.9 in across) is about 12.8 in³, so a soccer ball has room for about 27 baseballs' worth of space.
When the diameter is 3 times as large, the volume is \(3^3 = 27\) times as large. This "it grows with the cube" sense is the basis for estimating sizes of any kind, not just balls.
Estimating a serving of ice cream or food (cooking and food service)
If an ice cream scoop makes balls 2.5 in across, one scoop is a sphere with a radius of 1.25 in, about \(8.2\,\mathrm{in^3}\), or roughly 4.5 fluid ounces. Restaurants use calculations like this to control portion sizes and costs.
It works just as well for anything shaped into balls, such as meatballs, cookie dough and melon balls.
Finding the capacity of a spherical tank (industry and infrastructure)
Spherical storage tanks are round because a sphere holds the most volume for a given amount of material. A tank 60 ft across has a volume of \(\dfrac{4}{3}\pi \times 30^3 \approx 113{,}000\,\mathrm{ft^3}\), or about 846,000 gallons (1 ft³ ≈ 7.48 gallons).
The design and the stated capacity of such tanks are built on exactly the formula on this page.
Estimating the rough volume of a lump from a scan (medicine)
In CT or ultrasound scans, a roughly round lump or cyst is measured by its diameter, and its approximate volume can be estimated with the sphere formula. Medical imaging uses centimeters even in the US: a diameter of 2 cm gives about \(4.2\,\mathrm{cm^3}\), and 4 cm gives about \(33.5\,\mathrm{cm^3}\). Doubling the diameter makes the volume 8 times as large.
Real lumps are not perfect spheres, so this is only a rough estimate, used for example to follow whether something is growing.
Calculating the volume of the Earth, the Moon and other bodies (astronomy)
The Earth is nearly a sphere (strictly, it is slightly flattened), so with a radius of about 3,959 miles its volume is about \(\dfrac{4}{3}\pi \times 3959^3 \approx 2.60 \times 10^{11}\,\mathrm{mi^3}\), or 260 billion cubic miles.
The Moon's radius is about 1,080 miles, so its volume is only about \(\dfrac{1}{50}\) of the Earth's. The size comparisons of planets and moons in science textbooks are calculated with this formula.
Formulas and figures
Volume of a sphere (from the radius)
Figure
Standard notation
(the usual math form)
\(V\)
\(=\)
\(\dfrac{4}{3}\)
\(\times\)
\(\pi\)
\(\times\)
\(r\)
\(3\)
In words
(symbols replaced with words)
⑤
\(V\): volume of the sphere
\(=\)
④
\(\dfrac{4}{3}\): a constant
\(\times\)
③
\(\pi\): pi
\(\times\)
①
\(r\): radius
②
cubed (the number used as a factor 3 times)
The formula in words
①
Take the
\(r\): radius
②
and make it
cubed (the number used as a factor 3 times)
③
multiply by
\(\pi\): pi
④
multiply by
\(\dfrac{4}{3}\): a constant
⑤
and you get the
\(V\): volume of the sphere
Quick example
The volume of a sphere with a radius of 3 in is
\(V\): volume of the sphere
\(=\)
\(\dfrac{4}{3}\): a constant
\(\times\)
\(\pi\): pi
\(\times\)
radius (3 in)
cubed
\(\dfrac{4}{3} \times \pi \times 3^{3} = \dfrac{4}{3} \times \pi \times 27 = 36\pi\)
\(36\pi \approx 36 \times 3.14 = 113.04 \approx 113\,\mathrm{in^3}\)
Key idea
Why \(\dfrac{4}{3}\pi\)? The ancient Greek mathematician Archimedes found the answer. A cylinder that fits a sphere exactly (base radius \(r\), height \(2r\)) has a volume of \(\pi r^2 \times 2r = 2\pi r^3\), and the sphere takes up exactly \(\dfrac{2}{3}\) of it. So the sphere's volume is \(2\pi r^3 \times \dfrac{2}{3} = \dfrac{4}{3}\pi r^3\).
Say the formula out loud as "four-thirds pi r cubed" to remember it. The most common mistake is to multiply the radius by 3 instead of cubing it (\(r \times r \times r\)).
Volume of a sphere (from the diameter)
Figure
Standard notation
(the usual math form)
\(V\)
\(=\)
\(\pi\)
\(\times\)
\(d\)
\(3\)
\(\div\)
\(6\)
In words
(symbols replaced with words)
⑤
\(V\): volume of the sphere
\(=\)
①
\(\pi\): pi
\(\times\)
②
\(d\): diameter
③
cubed (the number used as a factor 3 times)
\(\div\)
④
\(6\): a constant
The formula in words
①
Take
\(\pi\): pi
②
and the
\(d\): diameter
③
multiply pi by the diameter
cubed (the number used as a factor 3 times)
④
divide by
\(6\): a constant
⑤
and you get the
\(V\): volume of the sphere
Quick example
The volume of a ball with a diameter of 6 in (radius 3 in) is
\(V\): volume of the sphere
\(=\)
\(\pi\): pi
\(\times\)
diameter (6 in)
cubed
\(\div\)
\(6\): a constant
\(\pi \times 6^{3} \div 6 = \pi \times 216 \div 6 = 36\pi\)
\(36\pi \approx 113\,\mathrm{in^3}\)
Key idea
This formula comes from putting "radius = half the diameter (\(r = d \div 2\))" into formula 1 and simplifying (\(\dfrac{4}{3}\pi \times \dfrac{d^3}{8} = \dfrac{\pi d^3}{6}\)). It is handy for things like balls, where you can measure the diameter but not the radius directly.
The 6 in diameter example gives the same answer (\(36\pi\)) as the 3 in radius example in formula 1, which shows that the two formulas describe the same sphere. Schools teach formula 1 (the radius formula), so learn that one first.
To find the volume of a sphere, cube the radius, then multiply by pi and by 4/3. If you only know the diameter, divide it by 2 to get the radius first. The answer is in the cube of the unit you entered (inches give in³), and dividing cubic inches by 231 gives US gallons.
Symbols and terms
Symbols
| \(V\) | vee | The usual symbol for volume, from the first letter of "volume". On this page it stands for the volume of the sphere. |
| \(r\) | ar | The radius of the sphere, from the first letter of "radius". It is the distance from the center of the sphere to its surface. |
| \(d\) | dee | The diameter of the sphere, from the first letter of "diameter". It is the distance straight across the widest part of the sphere, twice the radius (\(d = 2r\)). |
| \(\pi\) | pi | The ratio of a circle's circumference to its diameter. It goes on forever as about 3.14159…, and 3.14 is a common rounded value. |
| \(r^3\) | r cubed | The number \(r\) used as a factor 3 times (\(r \times r \times r\)). The small raised 3 is an exponent. (Example: \(3^3 = 3 \times 3 \times 3 = 27\)) |
| \(\mathrm{in^3}\) | cubic inches | A unit of volume. A cube that is 1 inch on each side has a volume of 1 in³. Do not mix it up with in² (square inches), which is a unit of area. |
| \(\mathrm{gal}\) | gallons | A unit for amounts of liquid such as water. 1 US gallon is \(231\,\mathrm{in^3}\), the size of a gallon jug of milk. |
| \(\mathrm{ft^3}\) | cubic feet | A unit of volume. A cube that is 1 foot on each side has a volume of 1 ft³. \(1\,\mathrm{ft^3} = 1728\,\mathrm{in^3} \approx 7.48\) gallons. |
Terms
| sphere | A perfectly round solid made of all the points that are the same distance from a center. Balls, marbles and soap bubbles are spheres; they look like a circle from any direction. |
| radius | The distance from the center of a sphere (or circle) to its surface. This is the value you enter in the calculator. It is half the diameter. |
| diameter | The length of a line from surface to surface through the center of a sphere (or circle). It is twice the radius. Balls are often described by their diameter. |
| volume | The amount of space a solid takes up, given as a number. It is measured by how many unit cubes, such as 1-inch cubes (1 in³), fit inside. |
| pi | The number of times a circle's diameter fits around its circumference (about 3.14), written \(\pi\). It appears not only in circle formulas but also in the volume and surface area of a sphere. |
| exponent | The small raised number in \(3^3\), which tells how many times the number is used as a factor. The volume formula has a power of 3 because a solid spreads out in 3 directions: length, width and height. |
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) |
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| What volume is and its units (Grade 5) |
|
| Multiplying a fraction by a whole number (Grades 4–5) |
|
| Exponents (Grades 6–8) |
|
| Spheres and the volume formula (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 of a sphere from the radius
| Radius r (in) | 5 |
| Volume (in³) | =4/3*PI()*B1^3 |
Table to find the volume of a sphere from the diameter
| Diameter d (in) | 10 |
| Volume (in³) | =PI()*B1^3/6 |
Table to convert cubic inches to US gallons
| Volume in in³ | 523.6 |
| Volume in US gallons | =B1/231 |
After pasting, column A holds the labels and column B holds the numbers. The upper row is your input, and the formula in the last row calculates automatically from it.
"PI()" is the Excel function that returns pi (3.14159…), "*" is multiplication, "/" is division and "^" is an exponent (how many times to multiply). "=4/3*PI()*B1^3" means "4/3 × π × B1 cubed".
In the first table, for example, B2 shows about 523.6 (in³ for a radius of 5 in). The second table is the same sphere (diameter 10 in), so it also shows about 523.6, and the third table shows about 2.267 (US gallons) in B2. Just replace the input numbers with your own.
"PI()" is the Excel function that returns pi (3.14159…), "*" is multiplication, "/" is division and "^" is an exponent (how many times to multiply). "=4/3*PI()*B1^3" means "4/3 × π × B1 cubed".
In the first table, for example, B2 shows about 523.6 (in³ for a radius of 5 in). The second table is the same sphere (diameter 10 in), so it also shows about 523.6, and the third table shows about 2.267 (US gallons) in B2. 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 of a sphere from the radius
| Radius r (in) | 5 |
| Volume (in³) | =4/3*PI()*B1^3 |
Table to find the volume of a sphere from the diameter
| Diameter d (in) | 10 |
| Volume (in³) | =PI()*B1^3/6 |
Table to convert cubic inches to US gallons
| Volume in in³ | 523.6 |
| Volume in US gallons | =B1/231 |
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 your own.
How to calculate it in Python
import math
radius = 5 # radius of the sphere (inches in this example)
volume = 4 / 3 * math.pi * radius ** 3 # volume of the sphere (input unit cubed, in3 in this example)
volume_gal = volume / 231 # in US gallons (for inputs in inches)
print(f"Volume: {volume} in3")
print(f"In US gallons: {volume_gal} gal")
Runs with the standard library only. "math.pi" is pi, "**" is an exponent (cubed here), "*" is multiplication and "/" is division. Change the radius at the top and run it (this example uses inches).
How to write it in LaTeX and other math languages (copy and paste)
Volume of a sphere (from the radius)
V = (4/3)πr³
V = \dfrac{4}{3}\pi r^{3}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>V</mi>
<mo>=</mo>
<mfrac><mn>4</mn><mn>3</mn></mfrac>
<mi>π</mi>
<msup><mi>r</mi><mn>3</mn></msup>
</mrow>
</math>
V = 4/3 pi r^3
(4/3)*Pi*r^3
V := (4/3)*Pi*r^3;
V = (4/3)*pi*r^3;
V = (4/3)πr^3
Volume of a sphere (from the diameter)
V = πd³/6
V = \dfrac{\pi d^{3}}{6}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>V</mi>
<mo>=</mo>
<mfrac>
<mrow><mi>π</mi><msup><mi>d</mi><mn>3</mn></msup></mrow>
<mn>6</mn>
</mfrac>
</mrow>
</math>
V = (pi d^3)/6
Pi*d^3/6
V := Pi*d^3/6;
V = pi*d^3/6;
V = πd^3/6
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 has a radius of 5 inches. Find each of the following: 1. The volume of the sphere in cubic inches (in³) (V = 4/3 × π × r³) 2. That volume in US gallons (1 gallon = 231 in³) Show the formulas you used and the numbers from the execution result.
How to Use
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1Enter your numbersType the numbers you want to calculate with into the input fields
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2CalculatePress the "Calculate" button
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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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Exponent Calculator (Powers, a to the nth Power)
Calculate powers (a to the nth power), such as the "radius cubed" in the volume of a sphere
Nth Root Calculator (Square Root, Cube Root, nth Root)
Calculate the cube root, which you need to work back from a volume to the radius
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