Enter the radius of the sphere. The surface area (4 × π × radius squared) is shown, along with the conversion to m².
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
-
What you can do on this page
-
What is this calculation used for?
-
How to Use
-
Formulas and figures
-
Symbols and terms
-
Good to know before you start
-
How to calculate it in Excel
-
How to calculate it in Google Sheets
-
How to calculate it in Python
-
How to write it in LaTeX and other math languages (copy and paste)
-
How to have ChatGPT do the calculation
-
DataChef Features
-
Related Features
-
NumberChef Calculators List
What you can do on this page
- Enter the radius of a sphere, and its surface area (surface area = \(4 \times \pi \times\) radius squared) 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 area in square feet (ft²) when you enter inches. Switch "Units" to Metric to use cm² and m² instead
- Also explains what to do when you only know the diameter (radius = diameter ÷ 2, or \(S = \pi d^2\))
- 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 surface area comes out in that unit squared (in² if you enter inches, ft² if you enter feet). For how much a sphere holds (its volume), see the sphere volume calculator page.
What is this calculation used for?
Estimating the paint for a spherical tank (industry and infrastructure)
Spherical storage tanks have their whole surface repainted from time to time, and the paint needed is estimated from the area to paint, the surface area of the sphere. For a tank 60 ft across, \(S = \pi d^2 = \pi \times 60^2 \approx 11{,}310\,\mathrm{ft^2}\). If a gallon covers about 350 ft², one coat takes about 33 gallons, and the numbers flow on into material costs and schedules.
Real tanks also have pipes and scaffolding, so the order starts from this value plus some extra.
Checking the Earth's surface area and "70% ocean" (geography and astronomy)
The Earth is nearly a sphere (strictly, it is slightly flattened), so with a radius of about 3,959 miles its surface area is \(4\pi \times 3959^2 \approx 1.97 \times 10^8\,\mathrm{mi^2}\), about 197 million square miles. The figures in science and geography books, "about 197 million square miles, about 70% of it ocean", come from estimates like this and from measurements.
For the Moon (radius about 1,080 miles), you get about 14.7 million square miles, only about \(\dfrac{1}{13}\) of the Earth's surface area.
Estimating the cover material for a ball (sports and manufacturing)
A size 5 soccer ball is about 8.7 in across, so its cover has an area of \(\pi \times 8.7^2 \approx 238\,\mathrm{in^2}\). Ball makers divide this surface area into panels (32 for the classic pentagons and hexagons) to work out how much leather or synthetic leather to cut.
This is the theoretical amount before adding seam allowances and waste, and the surface area formula is where it starts.
Body size and how fast an animal loses heat (science and biology)
If you treat an animal's body roughly as a sphere, halving the radius cuts the volume (a stand-in for body weight) to \(\dfrac{1}{8}\), but the surface area (the skin that loses heat) only to \(\dfrac{1}{4}\). So the smaller the body, the more surface area it has for each unit of volume, and the faster it loses heat.
Ideas such as "animals in cold regions tend to be larger" (Bergmann's rule) and "babies get cold faster than adults" are explained by this ratio of surface area to volume (real bodies are not spheres, so this is only a model for thinking).
Why water drops and soap bubbles are round (nature and chemistry)
It has been proved mathematically that, of all shapes that hold the same volume, the sphere has the smallest surface area. Water drops and soap bubbles are round because surface tension pulls their surface area down as far as it can.
On the other hand, things that should dissolve fast, like powdered milk or instant coffee, and things that should react fast, like the iron powder in disposable hand warmers, are made into fine grains to increase their surface area for each unit of volume. The surface area formula lets you check everyday effects like these with numbers.
Formulas and figures
Surface area of a sphere (from the radius)
Figure
Standard notation
(the usual math form)
\(S\)
\(=\)
\(4\)
\(\times\)
\(\pi\)
\(\times\)
\(r\)
\(2\)
In words
(symbols replaced with words)
⑤
\(S\): surface area of the sphere
\(=\)
④
\(4\): a constant
\(\times\)
③
\(\pi\): pi
\(\times\)
①
\(r\): radius
②
squared (the number times itself)
The formula in words
①
Take the
\(r\): radius
②
and make it
squared (the number times itself)
③
multiply by
\(\pi\): pi
④
multiply by
\(4\): a constant
⑤
and you get the
\(S\): surface area of the sphere
Quick example
The surface area of a sphere with a radius of 3 in is
\(S\): surface area of the sphere
\(=\)
\(4\): a constant
\(\times\)
\(\pi\): pi
\(\times\)
radius (3 in)
squared
\(4 \times \pi \times 3^{2} = 4 \times \pi \times 9 = 36\pi\)
\(36\pi \approx 36 \times 3.14 = 113.04 \approx 113\,\mathrm{in^2}\)
Key idea
Why multiply by \(4\pi\)? The ancient Greek mathematician Archimedes found the answer. Cut a sphere through its center, and the circle you see (the great circle) has an area of \(\pi r^2\). The surface area of the sphere is exactly 4 times that (\(\pi r^2 \times 4 = 4\pi r^2\)). Picture the skin of the sphere as enough to cover 4 of those circles.
Say the formula out loud as "four pi r squared" to remember it. It is easy to mix up with the volume, \(\dfrac{4}{3}\pi r^3\) (cubed). Think of the units: an area is length × length, so it is squared; a volume is length × length × length, so it is cubed.
Surface area of a sphere (from the diameter)
Figure
Standard notation
(the usual math form)
\(S\)
\(=\)
\(\pi\)
\(\times\)
\(d\)
\(2\)
In words
(symbols replaced with words)
④
\(S\): surface area of the sphere
\(=\)
①
\(\pi\): pi
\(\times\)
②
\(d\): diameter
③
squared (the number times itself)
The formula in words
①
Take
\(\pi\): pi
②
and the
\(d\): diameter
③
multiply pi by the diameter
squared (the number times itself)
④
and you get the
\(S\): surface area of the sphere
Quick example
The surface area of a ball with a diameter of 6 in (radius 3 in) is
\(S\): surface area of the sphere
\(=\)
\(\pi\): pi
\(\times\)
diameter (6 in)
squared
\(\pi \times 6^{2} = \pi \times 36 = 36\pi\)
\(36\pi \approx 113\,\mathrm{in^2}\)
Key idea
This formula comes from putting "radius = half the diameter (\(r = d \div 2\))" into formula 1 and simplifying (\(4\pi \times \dfrac{d^2}{4} = \pi d^2\)). It is handy for things like balls, where you can measure the diameter but not the radius directly.
You can also read it as "the surface area of a sphere is \(\pi\) times (about 3.14 times) the area of a square whose side is the diameter (\(d^2\))". 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 surface area of a sphere, square the radius, then multiply by pi and by 4. It is exactly 4 times the area of the great circle. If you only know the diameter, divide it by 2 to get the radius, or use S = πd². The answer is in the square of the unit you entered (inches give in²), and dividing square inches by 144 gives square feet.
Symbols and terms
Symbols
| \(S\) | ess | A common symbol for area and surface area, from the first letter of "surface". On this page it stands for the surface area of the sphere (some books write SA). |
| \(r\) | 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. |
| \(r^2\) | r squared | The number \(r\) multiplied by itself (\(r \times r\)). The small raised 2 is an exponent. (Example: \(3^2 = 3 \times 3 = 9\)) |
| \(\mathrm{in^2}\) | square inches | A unit of area. A square that is 1 inch on each side has an area of 1 in². Do not mix it up with in³ (cubic inches), which is a unit of volume. |
| \(\mathrm{ft^2}\) | square feet | A unit of area. A square that is 1 foot on each side has an area of 1 ft². \(1\,\mathrm{ft^2} = 144\,\mathrm{in^2}\) (the length factor 12, squared). |
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. |
| surface area | The total area of the outside of a solid. For a sphere, it is how much area the "skin" would cover if you could spread it all out. It is a different amount from the volume, which is how much the solid holds. |
| great circle | The largest circle you get when you cut a sphere with a flat plane through its center. Its radius is the same as the sphere's, and its area is \(\pi r^2\). The surface area of a sphere is exactly 4 times the area of its great circle. |
| 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 surface area and volume of a sphere. |
| exponent | The small raised number in \(3^2\), which tells how many times the number is used as a factor. The area formula has a power of 2 because a surface spreads out in 2 directions: length and width. |
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) |
|
| Area of a circle (Grade 7) |
|
| What area is and its units (Grades 3–5) |
|
| Exponents (Grades 6–8) |
|
| Spheres and the surface area formula (Grade 8 and high school geometry) |
|
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 sphere from the radius
| Radius r (in) | 5 |
| Surface area (in²) | =4*PI()*B1^2 |
Table to find the surface area of a sphere from the diameter
| Diameter d (in) | 10 |
| Surface area (in²) | =PI()*B1^2 |
Table to convert square inches to square feet
| Surface area in in² | 314.16 |
| Surface area in ft² | =B1/144 |
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*PI()*B1^2" means "4 × π × B1 squared".
In the first table, for example, B2 shows about 314.16 (in² for a radius of 5 in). The second table is the same sphere (diameter 10 in), so it also shows about 314.16, and the third table shows about 2.18 (ft²) 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*PI()*B1^2" means "4 × π × B1 squared".
In the first table, for example, B2 shows about 314.16 (in² for a radius of 5 in). The second table is the same sphere (diameter 10 in), so it also shows about 314.16, and the third table shows about 2.18 (ft²) 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 surface area of a sphere from the radius
| Radius r (in) | 5 |
| Surface area (in²) | =4*PI()*B1^2 |
Table to find the surface area of a sphere from the diameter
| Diameter d (in) | 10 |
| Surface area (in²) | =PI()*B1^2 |
Table to convert square inches to square feet
| Surface area in in² | 314.16 |
| 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 your own.
How to calculate it in Python
import math
radius = 5 # radius of the sphere (inches in this example)
surface_area = 4 * math.pi * radius ** 2 # surface area of the sphere (input unit squared, in2 in this example)
surface_area_ft2 = surface_area / 144 # in square feet (for inputs in inches)
print(f"Surface area: {surface_area} in2")
print(f"In square feet: {surface_area_ft2} ft2")
Runs with the standard library only. "math.pi" is pi, "**" is an exponent (squared 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)
Surface area of a sphere (from the radius)
S = 4πr²
S = 4\pi r^{2}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>S</mi>
<mo>=</mo>
<mn>4</mn>
<mi>π</mi>
<msup><mi>r</mi><mn>2</mn></msup>
</mrow>
</math>
S = 4 pi r^2
4*Pi*r^2
S := 4*Pi*r^2;
S = 4*pi*r^2;
S = 4πr^2
Surface area of a sphere (from the diameter)
S = πd²
S = \pi d^{2}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>S</mi>
<mo>=</mo>
<mi>π</mi>
<msup><mi>d</mi><mn>2</mn></msup>
</mrow>
</math>
S = pi d^2
Pi*d^2
S := Pi*d^2;
S = pi*d^2;
S = πd^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 has a radius of 5 inches. Find each of the following: 1. The surface area of the sphere in square inches (in²) (S = 4 × π × r²) 2. That surface area in square feet (1 ft² = 144 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
DataChef Features
Easy and Free
Unlimited conversions for free.
No technical knowledge required.
Intuitive and user-friendly operation.
No technical knowledge required.
Intuitive and user-friendly operation.
No Registration Required
Available immediately after access.
Can be used without registering personal information.
Can be used without registering personal information.
Safe and Secure
Fully SSL encrypted communication.
Automatic file deletion by clicking "download".
Automatic file deletion by clicking "download".
Fast
High-speed site access
and rapid file conversion.
and rapid file conversion.
No Watermark
No watermark.
No attribution required.
No attribution required.
Commercial Use Available
Free for commercial use.
No need to contact us for commercial use permission.
No need to contact us for commercial use permission.
Related Features
Sphere Volume Calculator (V = 4/3 × π × r³)
Find how much the same sphere holds (its volume). A good formula to learn together with surface area
Circle Calculator (Radius, Diameter, Circumference and Area)
For the circle you get when you slice a sphere, find the radius, diameter, circumference and area all at once
Nth Root Calculator (Square Root, Cube Root, nth Root)
Calculate the square root, which you need to work back from a surface area to the radius
Ad 
