The space diagonal runs from one corner of a box to the opposite corner, straight through the interior — the longest straight line that fits inside. It comes from the Pythagorean theorem used twice: first across the base to get a face diagonal of √(l² + w²), then up to the far corner, giving d = √(l² + w² + h²). For a cube every dimension equals s, so the space diagonal simplifies to s√3. This is a favourite GRE twist because it looks three-dimensional but is really the same triple-friendly Pythagoras you already know — a 3-4-12 box, for instance, has a space diagonal of exactly 13. Don’t confuse the space diagonal with a face diagonal (which uses only two dimensions); read which corner-to-corner distance the question wants.
✅ Solved examples
1. A box measures 3 by 4 by 12. Find its space diagonal.
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