Geometry
How To Calculate The Surface Area Of A Sphere
A sphere's surface area is exactly four of its great circles, a result Archimedes was so pleased with that he asked for it on his tombstone.
Quick Answer
surface area = 4 x pi x radius^2
- A
- Surface area of the sphere, in square units
- r
- Radius, the distance from centre to surface
Square the radius, multiply by pi, then multiply by four. A sphere of radius five has a surface area of about 314.16 square units, which is the same as four flat circles of that radius laid side by side.
What Is The Surface Area Of A Sphere?
The surface area of a sphere is the total area of its curved skin, given by four times pi times the radius squared. A ball of radius five has a surface area of about 314.16 square units.
The four is the memorable part. It means the surface of a sphere has exactly the same area as four flat circles of the same radius, which is a genuinely surprising result and the reason Archimedes considered it one of his finest achievements.
A related fact falls out of the same work: the sphere occupies exactly two thirds of the volume of the smallest cylinder that contains it, curved surface included. Both results come from the same comparison with the enclosing cylinder.
The radius is the distance from the centre to any point on the surface, and it is half the diameter. Since the radius is squared, using the diameter by mistake multiplies the answer by four, the same trap as with circles and cones.
Surface area grows with the square of the radius while volume grows with the cube. Doubling the radius quadruples the surface but multiplies the volume by eight. This is why large objects lose heat more slowly than small ones: they have relatively little surface for their bulk.
The great circle is the largest circle you can draw on a sphere, a slice through the centre. Its area is pi times the radius squared, so the sphere's surface is exactly four great circles. On a globe, the equator is a great circle.
The surface area to volume ratio is the surface area divided by the volume, which simplifies to three over the radius. It falls as the sphere gets bigger, and it drives everything from why elephants have big ears to why nanoparticles are chemically reactive.
Real spheres are rarely perfect. A ball bearing is close, a planet is not quite, and a bubble wobbles. For everyday work the ideal formula is accurate enough, but precision engineering on a manufactured sphere accounts for sphericity tolerance.
Units behave as they do for any area. A radius in centimetres gives a surface area in square centimetres. The radius must be squared and the result is a two dimensional quantity, never a volume.
Spheres appear everywhere once you start looking: ball bearings, bubbles, droplets, planets, and the approximation of a raindrop. Surface area is the quantity that governs evaporation, drag, heat loss and coating costs.
Finally, note that a hemisphere is exactly half the surface, but only if you mean the curved part alone. A dome including its flat circular base has an extra pi times radius squared to add on, making three times pi times radius squared in total.
Formula
A = 4 x pi x r^2
Four great circles. The radius must be squared, and the result is in square units.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| A | Surface area | square metres | Total area of the curved surface. |
| r | Radius | metres | Centre to surface distance. |
V = 4 x pi x r^3 / 3
The companion formula. Note the cube here against the square in the surface area.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| V | Volume | cubic metres | Space inside the sphere. |
| r | Radius | metres | Centre to surface distance. |
A / V = 3 / r
Falls as the sphere grows, which is why large bodies retain heat and small ones lose it quickly.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| A | Surface area | square metres | Outside area. |
| V | Volume | cubic metres | Enclosed space. |
| r | Radius | metres | Centre to surface distance. |
How To Calculate The Surface Area Of A Sphere
- 1
Find the radius
Halve the diameter if that is what you measured. The radius is what gets squared, so using the diameter makes the answer four times too large.
- 2
Square the radius
A radius of five squares to twenty five. Squaring is the step that turns a length into an area, which is why the unit becomes square.
- 3
Multiply by pi
Twenty five times pi is about 78.54, which is the area of the great circle, the largest flat slice through the centre.
- 4
Multiply by four
78.54 times four is about 314.16. The sphere's skin is exactly four of those great circles, no more and no less.
- 5
Keep the unit squared
Centimetres give square centimetres. If a volume is wanted instead, the formula changes to four thirds pi r cubed and the unit becomes cubic.
Examples
Example 1: A ball of radius five
- Radius
- 5
| Step | Calculation | Result |
|---|---|---|
| Great circle area | pi x 5^2 | 78.5398 |
| Volume | 4 x pi x 125 / 3 | 523.5988 |
| Surface area | 4 x 78.5398 | 314.1593 |
Result: A ball of radius five has a surface area of 314.1593 square units, exactly four times the area of its great circle.
Example 2: A football with a radius of eleven centimetres
- Radius
- 11 cm
| Step | Calculation | Result |
|---|---|---|
| Great circle area | pi x 11^2 | 380.1327 |
| Volume | 4 x pi x 1331 / 3 | 5575.2798 |
| Surface area | 4 x 380.1327 | 1520.5308 |
Result: A football of radius eleven centimetres has 1520.5308 square centimetres of surface, which is roughly how much leather goes into the panels.
Example 3: A water droplet of radius 0.5 millimetres
- Radius
- 0.5 mm
| Step | Calculation | Result |
|---|---|---|
| Great circle area | pi x 0.5^2 | 0.7854 |
| Surface to volume ratio | 3 / 0.5 | 6 |
| Surface area | 4 x 0.7854 | 3.1416 |
Result: A droplet of radius 0.5 millimetres has a surface area of 3.1416 square millimetres and a surface to volume ratio of six, which is why fine sprays evaporate so quickly.
Calculator
Surface area
314.1593
- Volume
- 523.5988
- Great circle area
- 78.5398
- Surface area to volume ratio
- 0.6
Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.
Prefer a full-width tool? Open the The Surface Area Of A Sphere calculator page.
Common Mistakes
Using the diameter as the radius
The radius is half the diameter and it is the radius that gets squared. Substituting the diameter multiplies the surface area by four, and the answer still looks reasonable.
Using the volume formula
Surface area uses the radius squared, volume uses it cubed. Both start with four pi, which makes them easy to confuse, but they answer completely different questions.
Dropping the four
The four is not decorative. It is the number of great circles that fit on the sphere, and leaving it out gives a quarter of the correct area.
Writing the answer in cubic units
Surface area is two dimensional. Square centimetres is right, cubic centimetres describes the volume instead.
Treating a hemisphere as half the surface with a base
Half the sphere's curved surface is two pi r squared. If the flat circular base is also part of the object, add pi r squared for a total of three pi r squared.
Assuming the surface to volume ratio stays constant
It falls as three over the radius, so bigger spheres have proportionally less surface. Ignoring that leads to wrong conclusions about cooling, evaporation and reaction rates.
Applying the formula to an ellipsoid
A squashed sphere is an ellipsoid with a much more complicated area formula. The sphere formula is only exact when every radius is the same.
FAQ
What is the formula for the surface area of a sphere?
Four times pi times the radius squared. Square the radius, multiply by pi, then multiply by four, and the answer is in square units.
Why is the surface area four times a circle?
Because a sphere's skin has exactly the same area as four great circles of the same radius. It is not obvious, but it follows from comparing the sphere with the cylinder that encloses it.
How is surface area different from volume?
Surface area measures the outside skin in square units, while volume measures the space inside in cubic units. The sphere's surface area uses the radius squared and its volume uses the radius cubed.
What is a great circle?
Any circle on the sphere whose plane passes through the centre, such as the equator on a globe. It is the largest circle you can draw on the surface, with an area of pi times the radius squared.
What happens to the surface area if the radius doubles?
It quadruples. Because the radius is squared, a factor of two in the radius becomes a factor of four in the area, while the volume grows by a factor of eight.
How do you find the surface area of a hemisphere?
Half the sphere gives two pi r squared for the curved part. If the flat base counts too, add pi r squared, making three pi r squared in total.
References
- [1]Wikipedia, Sphere — https://en.wikipedia.org/wiki/Sphere
- [2]Wikipedia, Surface area — https://en.wikipedia.org/wiki/Surface_area
- [3]Math is Fun, Sphere — https://www.mathsisfun.com/geometry/sphere.html
- [4]Wikipedia, On the Sphere and Cylinder — https://en.wikipedia.org/wiki/On_the_Sphere_and_Cylinder