3D Box Diagonal Calculator
This 3D box diagonal calculator finds the space diagonal of a rectangular box (cuboid) from its length, width, and height, along with the three face diagonals. The space diagonal is the longest straight line that fits inside the box, running corner to opposite corner, and is found by extending the Pythagorean theorem into three dimensions.
Formula
Space: √(l²+w²+h²); Face LW: √(l²+w²); Face LH: √(l²+h²); Face WH: √(w²+h²)
- The space diagonal connects two opposite corners through the interior of the box: d = √(l² + w² + h²).
- Each face is a rectangle with its own diagonal: √(l² + w²), √(l² + h²), and √(w² + h²).
- The formula is the Pythagorean theorem applied twice - first across a face, then up to the opposite corner.
- The space diagonal is always longer than any edge or face diagonal of the same box.
- All three dimensions must use the same unit; the diagonals come out in that same unit.
Example Calculation
Inputs
- Length: 10 cm
- Width: 8 cm
- Height: 6 cm
For a box 10 × 8 × 6 cm, the space diagonal is √(10² + 8² + 6²) = √200 ≈ 14.14 cm. The face diagonals are √(10² + 8²) ≈ 12.81 cm, √(10² + 6²) ≈ 11.66 cm, and √(8² + 6²) = 10 cm.
Frequently asked questions
What is the space diagonal of a box?
It is the longest straight line inside a rectangular box, running from one corner to the opposite corner through the interior. It equals √(l² + w² + h²).
How is the space diagonal formula derived?
By applying the Pythagorean theorem twice: first to a face to get √(l² + w²), then combining that with the height to get √(l² + w² + h²).
What is the difference between a space diagonal and a face diagonal?
A face diagonal crosses one rectangular face using two dimensions, while the space diagonal cuts through the inside of the box using all three dimensions.
Will the longest object fit inside the box?
The space diagonal is the longest straight item that can fit inside, so an object fits corner-to-corner if its length is no greater than √(l² + w² + h²).
Does it matter which dimension is length, width, or height?
No. The space diagonal √(l² + w² + h²) is symmetric, so swapping the dimensions gives the same result. Only the face-diagonal labels change.
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