Case 1
- Length
- 3
- Width
- 4
- Height
- 12
13.000000
Open with these valuesBox Diagonal Calculator
13.000000units
Result: 13.000000 unitsThe longest straight line that fits inside a box runs corner to opposite corner: √(l² + w² + h²). It is Pythagoras used twice, once across the floor and once up the side. All three edges go in with the same length unit, and the diagonal comes back in it.
The numbers at a glance
Held fixed: Length 3.000, Width 4.000.
| Height | Result |
|---|---|
| 5.000 | 7.071068 |
| 10.000 | 11.180340 |
| 12.000Your value | 13.000000 |
| 15.000 | 15.811388 |
| 20.000 | 20.615528 |
Worked examples
Case 2
- Length
- 2
- Width
- 3
- Height
- 4
5.385165
Open with these valuesCase 3
- Length
- 5
- Width
- 5
- Height
- 5
8.660254
Open with these valuesHow it's calculated
d = √(l² + w² + h²)
- StepMeasure the three edges that meet at one corner.
- StepEnter all three in the same length unit.
- ResultRead the space diagonal in that same unit.
Reference table
| Length, width, height | Exact | Diagonal |
|---|---|---|
| 1, 1, 1 | √3 | 1.732051 |
| 0.5, 2, 3 | √13.25 | 3.640055 |
| 2, 3, 4 | √29 | 5.385165 |
| 5, 5, 5 | 5√3 | 8.660254 |
| 3, 4, 12 | 13 | 13.000000 |
Questions
How do you find the diagonal of a box?
Square the three edge lengths, add them up and take the square root. A 3 by 4 by 12 box has a diagonal of exactly 13.
Why is it Pythagoras twice?
The floor diagonal is √(l² + w²). That diagonal and the height form a right triangle, and its hypotenuse is the space diagonal.
Will a rod of that length fit in the box?
Only if the rod is thin. The space diagonal is the longest line that fits, so anything longer cannot go in at any angle.
Does the order of the three edges matter?
No. All three are squared and added, so length, width and height are interchangeable.
Sources and last check
Information, not professional advice.