A cube is fixed entirely by one edge, so its volume is the edge cubed and its surface area is six times the edge squared. The cubing is what makes intuition fail: an edge only 26% longer doubles the volume, because 1.26 cubed is very close to two. That is the reason storage capacity grows so much faster than a container looks like it should.
The surface-to-volume ratio falls as a cube grows, at exactly six divided by the edge. Small cubes have proportionally enormous surface area, which is why crushed ice melts faster than a block and why small animals lose heat so much more quickly than large ones. The space diagonal, edge times root three, is the longest straight line that fits inside.
The formula
s- Edge length
V- Volume
A- Total surface area of all six faces
d- Space diagonal, corner to opposite corner
How it works, step by step
- Enter the edge length of the cube.
- The gauge shows the volume.
- Surface area and both diagonals appear below.
- The last figure is the edge you would need to double the volume.
Worked examples
An edge of 6
Volume is 6 cubed = 216 and surface area is 6 × 36 = 216. The space diagonal is 6 × 1.7320508 = 10.3923, the longest rod that fits inside.
Doubling the volume
To reach 432 cubic units the edge must be the cube root of 432, which is 7.5595. Only a 25.99% increase in edge doubles the capacity.
How to read your score
Frequently asked questions
Why does the volume grow so fast?
Because all three dimensions grow together. A 10% longer edge gives 1.1 cubed, or 33% more volume.
What is the surface-to-volume ratio for?
It predicts how quickly something exchanges heat or moisture with its surroundings. For a cube it is six divided by the edge, so it falls as the cube grows.
How do I find the edge from the volume?
Take the cube root. A volume of 343 cubic units means an edge of 7.
What is the difference between the face and space diagonals?
The face diagonal crosses one square face and is the edge times root two. The space diagonal runs through the interior between opposite corners and is the edge times root three.
Cube properties by edge length
| Edge | Volume | Surface area | Space diagonal | Surface per unit volume |
|---|---|---|---|---|
| 1 | 1 | 6 | 1.7321 | 6.0000 |
| 2 | 8 | 24 | 3.4641 | 3.0000 |
| 3 | 27 | 54 | 5.1962 | 2.0000 |
| 4 | 64 | 96 | 6.9282 | 1.5000 |
| 5 | 125 | 150 | 8.6603 | 1.2000 |
| 6 | 216 | 216 | 10.3923 | 1.0000 |
| 8 | 512 | 384 | 13.8564 | 0.7500 |
| 10 | 1000 | 600 | 17.3205 | 0.6000 |
| 15 | 3375 | 1350 | 25.9808 | 0.4000 |
| 20 | 8000 | 2400 | 34.6410 | 0.3000 |
The final column shows the surface-to-volume ratio falling as the cube grows.