The space diagonal of a box is the root of the sum of the squares of all three dimensions, and it is the longest rigid straight object that will fit inside. It is Pythagoras applied twice: once across the base to get the face diagonal, then again from that diagonal up to the opposite corner.
For packing, this is the number that actually matters, and it is always noticeably longer than the longest side. A 60 by 40 by 40 carton takes an 82.5 cm rod even though no side exceeds 60 cm. The caveat is that the item must be genuinely thin: a thick object cannot follow the true diagonal because its own girth needs room in the corners.
The formula
l, w, h- The three internal dimensions
d- Space diagonal
Face diagonals- The three two-dimensional diagonals across each pair of faces
How it works, step by step
- Enter the three internal dimensions of the box.
- Enter the length of the item you want to pack.
- The gauge shows the space diagonal.
- The clearance figure tells you whether the item fits.
Worked examples
A 60 by 40 by 40 cm carton
The space diagonal is the root of 3600 + 1600 + 1600 = 82.4621 cm, so a 75 cm rod fits with 7.462 cm to spare — despite no side reaching 75 cm.
A flat box
A 100 by 70 by 5 cm box has a space diagonal of 122.168 cm, barely more than the 122.066 cm base diagonal. When one dimension is small it contributes almost nothing.
How to read your score
Frequently asked questions
Why is the diagonal longer than the longest side?
Because it uses all three dimensions at once. Adding any depth to a diagonal path always lengthens it, which is why even a shallow box gains a little reach.
Can I always use the full diagonal in practice?
Only for genuinely thin items. A thick object needs clearance for its own girth at both corners, so the usable length is less than the geometric diagonal.
Should I use internal or external dimensions?
Internal. External measurements include the wall thickness, which is not available for packing.
How does this relate to the flat diagonal?
The face diagonal is the same calculation with one dimension set to zero. The space diagonal is the face diagonal combined with the third dimension by Pythagoras again.
Space diagonals of common cartons
| Dimensions | Longest side | Base diagonal | Space diagonal | Gain over longest side |
|---|---|---|---|---|
| 30 x 20 x 15 | 30 | 36.06 | 39.05 | 30.2% |
| 40 x 30 x 20 | 40 | 50.00 | 53.85 | 34.6% |
| 60 x 40 x 40 | 60 | 72.11 | 82.46 | 37.4% |
| 50 x 50 x 50 | 50 | 70.71 | 86.60 | 73.2% |
| 100 x 70 x 5 | 100 | 122.07 | 122.17 | 22.2% |
| 120 x 80 x 80 | 120 | 144.22 | 164.92 | 37.4% |
The final column shows how much extra reach the diagonal buys over the longest side.