Box Space Diagonal Calculator
Find the space diagonal of a rectangular box (cuboid) from its length, width, and height using the 3D Pythagorean theorem. Enter three dimensions to compute the longest internal diagonal.
How to use this tool
- Enter length, width and height in the fields above.
- Results update instantly as you type — or click Calculate.
- Read your space diagonal and the full breakdown beneath it.
Formula
d = sqrt(l^2 + w^2 + h^2)
How it works
Extend the Pythagorean theorem to 3D: d = √(l² + w² + h²).
Worked example
3-4-12 box gives diagonal 13
- d
- =
- s
- q
- r
- t
- (
- 9
- +
- 1
- 6
- +
- 1
- 4
- 4
- )
- =
- s
- q
- r
- t
- (
- 1
- 6
- 9
- )
- =
- 1
- 3
- .
Common mistakes to avoid
- Using the 2D formula sqrt(l squared + w squared) and forgetting to add h squared — the space diagonal of a 3D box requires all three dimensions.
- Confusing a face diagonal (diagonally across one rectangular face) with the space diagonal (corner to corner through the interior of the box).
- Entering dimensions in inconsistent units — for example length in inches and width in feet — which produces a meaningless result.
Key terms
Frequently asked questions
- What is the difference between a face diagonal and a space diagonal?
- A face diagonal crosses one flat rectangular face of the box using only two dimensions. The space diagonal passes through the interior of the box from one corner to the opposite corner and uses all three dimensions.
- Can I use this to find whether a long object fits inside a box?
- Yes — the space diagonal is the longest straight line that fits inside a rectangular box. If your object is shorter than the space diagonal, it can fit diagonally.
- Is the formula the same for a cube?
- Yes. For a cube with side s, d = sqrt(s squared + s squared + s squared) = s x sqrt(3). It is a special case of the general formula.