The longest straight line that fits inside a rectangular box is its space diagonal: the line from one corner to the opposite corner. Calculating it requires extending Pythagoras's two-dimensional theorem into three dimensions — a small but elegant generalization that solves real-world problems.
The formula
d = √(l² + w² + h²)
For a box with length 5, width 3, and height 4:
d = √(25 + 9 + 16) = √50 ≈ 7.07
Whatever unit you measured in, that's the unit of the diagonal.
Where the formula comes from
Start with the standard Pythagorean theorem. For a right triangle with legs a and b and hypotenuse c:
c² = a² + b²
Now consider a rectangular box. Look at the bottom face — a rectangle with sides l and w. The diagonal of this bottom face is:
d_floor = √(l² + w²)
Now imagine the space diagonal: it goes from one bottom corner up to the opposite top corner. This forms the hypotenuse of a right triangle where:
- One leg is the floor diagonal: √(l² + w²)
- The other leg is the height: h
Apply Pythagoras again:
d_space² = d_floor² + h² d_space² = (l² + w²) + h² d_space = √(l² + w² + h²)
Pythagoras applied twice. The result generalizes: in n dimensions, the diagonal of a rectangular box is the square root of the sum of squared sides.
Worked examples
A standard moving box
An 18 × 14 × 10 inch moving box:
d = √(18² + 14² + 10²) = √(324 + 196 + 100) = √620 ≈ 24.9 inches
You can fit something just under 25 inches long, as long as it's thin enough to angle from corner to corner.
A 40-foot shipping container
40 ft long, 8 ft wide, 8.5 ft tall:
d = √(40² + 8² + 8.5²) = √1,736.25 ≈ 41.7 feet
So a 41-foot pole can fit diagonally in a 40-foot container.
A coffin
Roughly 84 × 28 × 23 inches:
d = √(84² + 28² + 23²) = √8,369 ≈ 91.5 inches
The kind of math undertakers actually do.
Face diagonal vs space diagonal
A rectangular box has two kinds of diagonals:
- Face diagonals are the diagonals across each rectangular face. Six in total.
- Space diagonals go from corner to opposite corner through the interior. Four in total, all the same length.
| Diagonal type | Formula | For a 5 × 3 × 4 box |
|---|---|---|
| Face diagonal (l × w face) | √(l² + w²) | √34 ≈ 5.83 |
| Face diagonal (l × h face) | √(l² + h²) | √41 ≈ 6.40 |
| Face diagonal (w × h face) | √(w² + h²) | √25 = 5.00 |
| Space diagonal | √(l² + w² + h²) | √50 ≈ 7.07 |
The space diagonal is always longer than any face diagonal.
Practical application: will it fit?
- Measure the item's length.
- Calculate the box's space diagonal.
- If the diagonal is longer than the item, it fits (assuming the item is thin enough to angle through).
A 22-inch sword in an 18 × 12 × 8 box?
Diagonal = √(324 + 144 + 64) = √532 ≈ 23.1 inches
Yes — 22 fits in 23.1.
Doorways: a different kind of diagonal problem
For getting furniture through a doorway, the diagonal you care about is the face diagonal:
Face diagonal = √(width² + depth²)
Compared to the doorway's diagonal:
Doorway diagonal = √(doorway width² + doorway height²)
If the appliance's face diagonal is less than the doorway's diagonal, you can tilt-walk it through. This is a 2D problem, not 3D.
Higher dimensions
The formula generalizes to any number of dimensions. For an n-dimensional rectangular "box" with sides s₁, s₂, ..., sₙ:
d = √(s₁² + s₂² + ... + sₙ²)
In one dimension, just the length of a line segment. In two, standard Pythagoras. In three, the box's space diagonal. In four (a "tesseract"), the formula still works.
Common mistakes
Adding instead of squaring
Using l + w + h instead of √(l² + w² + h²). For a 5 × 3 × 4 box, the sum is 12 — much larger than the actual diagonal of 7.07.
Forgetting the square root
Half-correct: l² + w² + h². For our box, that's 50. The diagonal is √50, not 50.
Mixing units
Convert all to the same unit first.
The diagonal of a cube
For a cube with side s:
d = √(3s²) = s × √3 ≈ 1.732 × s
For a 10 cm cube, the diagonal is 17.32 cm. √3 (Theodorus's constant) shows up in 3D geometry the way √2 shows up in 2D.
The takeaway
The space diagonal formula — d = √(l² + w² + h²) — is the 3D extension of Pythagoras's theorem. It tells you the longest straight line that fits inside a rectangular box. The math is straightforward: square each side, add them up, take the square root.