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Cuboid Space Diagonal Calculator

Find the longest diagonal inside any box

How it is calculated

The space diagonal runs from one bottom corner to the opposite top corner through the inside of the box: d = √(l² + w² + h²), where l, w and h are the length, width and height. It comes from applying Pythagoras twice: first the floor diagonal √(l² + w²), then that diagonal with the height. All three sides must be above zero. The tool does not convert units; it simply adds the label cm to the answer, so enter the sides in centimetres for the label to be right (any other unit gives a number in that same unit). The result is rounded to two decimals. The value is for the inside dimensions: wall thickness, padding and the object's own width are not considered.

Worked example

A box 120 cm long, 40 cm wide and 30 cm high. Squares: 120² = 14,400, 40² = 1,600, 30² = 900. Sum: 16,900. Square root: 130.00 cm. So a straight rod up to 130 cm long could fit corner to corner, even though the longest side is only 120 cm.

Try this example →

Common mistakes

  • Measuring the outside of the box and expecting the inside diagonal. Cardboard walls and padding reduce the usable space.
  • Mixing units, for example length in metres and height in centimetres. All three sides must be in one unit, and the output is labelled cm.
  • Believing a pole as long as the diagonal fits easily. It only fits when it lies exactly corner to corner, with no room for the thickness of the object.
  • Using only the floor diagonal. It leaves out the height and gives a shorter figure than the true space diagonal.

About this calculator

The space diagonal of a rectangular box follows directly from the 3D Pythagorean theorem: d = √(l² + w² + h²) — apply the theorem once across the base to get the floor diagonal, then again between that diagonal and the height to get the true corner-to-corner distance through the box's interior.

Movers and shippers use it to check whether a long item like a pole, mirror, or flat-pack furniture piece will actually fit diagonally inside a box or truck bed that's too short for it to lie flat, and furniture makers and packaging engineers use the same formula when sizing crates around irregular or oversized objects.

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Comments (1)

  • Editorial team

    Whether a long item fits depends on how it can be tilted. For the 120 x 40 x 30 cm box in the guide, the 130 cm diagonal is only reachable if the item rises from one bottom corner to the opposite top corner. If it has to lie flat, the limit is the floor diagonal, the square root of 120 squared plus 40 squared, about 126.49 cm. What are you trying to fit, and how tall is it across?

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