A perfect cuboid is a cuboid having integer side lengths, integer face diagonals d_(a b) | = | sqrt(a^2 + b^2) d_(a c) | = | sqrt(a^2 + c^2) d_(b c) | = | sqrt(b^2 + c^2), and an integer space diagonal d_(a b c) = sqrt(a^2 + b^2 + c^2). The problem of finding such a cuboid is also called the brick problem, diagonals problem, perfect box problem, perfect cuboid problem, or rational cuboid problem. No perfect cuboids are known despite an exhaustive search for all "odd sides" up to 10^10 (Butler, pers. comm., Dec. 23, 2004).

cuboid | Diophantine equation | Euler brick | face diagonal | Heronian tetrahedron | Heronian triangle | integer triangle | space diagonal