A sphere holds four thirds pi r cubed and has a surface area of four pi r squared. The neat relationship between them is that the surface area is exactly the derivative of the volume with respect to radius, which makes physical sense: growing a sphere by a thin shell adds a volume equal to the surface area times the shell thickness.
A sphere fills pi over six of the cube that just contains it, which is 52.36%. It also has the smallest possible surface area for a given volume, which is why droplets, bubbles and planets pull themselves into spheres. Any other shape holding the same volume must expose more surface, and surface costs energy.
The formula
r- Radius
V- Volume
A- Surface area
πr<sup>2</sup>- Great-circle area, the shadow the sphere casts
How it works, step by step
- Enter the radius. Halve the diameter first if that is what you measured.
- The gauge shows the volume.
- Surface area and great-circle area appear below.
- The final figure shows how much of the enclosing cube the sphere fills.
Worked examples
A radius of 5
Volume is four thirds pi × 125 = 523.5988 and surface area is 4 pi × 25 = 314.1593. The enclosing 10 by 10 by 10 cube holds 1000, so the sphere fills 52.36% of it.
Doubling the volume
The radius must rise by the cube root of two, from 5 to 6.2996. Surface area grows only by the cube root of four, so it rises from 314.16 to 498.70.
How to read your score
Frequently asked questions
Why is the surface area formula four pi r squared?
It is exactly four times the great-circle area, a result Archimedes proved by comparing the sphere with its circumscribed cylinder. It is also the derivative of the volume formula.
How much of a box does a ball fill?
Pi over six, about 52.36%, when the ball just touches all six faces. Nearly half the box is empty space, which is why spherical items ship badly.
Why are bubbles spherical?
Because a sphere minimises surface area for a fixed volume, and surface tension makes surface area energetically expensive. Any deviation costs energy.
How do I find the radius from the volume?
Multiply the volume by three, divide by four pi, then take the cube root. A volume of 500 gives a radius of about 4.924.
Sphere volume and area by radius
| Radius | Volume | Surface area | Great-circle area | Surface per unit volume |
|---|---|---|---|---|
| 0.5 | 0.5236 | 3.1416 | 0.7854 | 6.0000 |
| 1 | 4.1888 | 12.5664 | 3.1416 | 3.0000 |
| 2 | 33.5103 | 50.2655 | 12.5664 | 1.5000 |
| 3 | 113.0973 | 113.0973 | 28.2743 | 1.0000 |
| 5 | 523.5988 | 314.1593 | 78.5398 | 0.6000 |
| 7 | 1436.7550 | 615.7522 | 153.9380 | 0.4286 |
| 10 | 4188.7902 | 1256.6371 | 314.1593 | 0.3000 |
| 15 | 14137.1669 | 2827.4334 | 706.8583 | 0.2000 |
| 25 | 65449.8469 | 7853.9816 | 1963.4954 | 0.1200 |
Volume grows with the cube of the radius while surface area grows only with its square.