Space diagonal of a rectangular box

For edges a, b, c the longest stick inside is √(a²+b²+c²).

Key formulas

Face diagonal on the base is √(a²+b²). Combine with height c using Pythagoras again.

3, 4, 12 → d = √(9+16+144)=√169=13. A neat Pythagorean triple in three dimensions.

“Will a rod of length L fit?” Compare L with d. Orientation can rotate; the space diagonal is the maximum.

Problem. Will a rod of length 13 cm fit inside a 3 cm × 4 cm × 12 cm box?

Solution. Space diagonal d = √(3²+4²+12²) = √(9+16+144) = √169 = 13. Yes — it fits exactly along the space diagonal.

d=√(4+16+16)=√36=6.

Box 2×3×6. Space diagonal is:

Across one face uses two edges; through the interior uses three.