Pythagoras in three dimensions

Pythagoras applies to a right-angled triangle: the square of its hypotenuse equals the sum of the squares of its other sides. In a cuboid with perpendicular edge lengths a, b and c, the base diagonal has square a² + b². That diagonal and the vertical edge form another right-angled triangle, giving a space diagonal of √(a² + b² + c²).

Draw or identify the triangle containing the required length. A sloping edge in a pyramid may need the distance from the base centre to a corner, rather than half one base side. Keep exact square roots until the final rounding. Every length in the calculation must use the same unit, and the final answer needs a length unit.

Worked example

A cuboid has edges 3 cm, 4 cm and 12 cm. Find its space diagonal.

1. The base diagonal is √(3² + 4²) = 5 cm.
2. The space diagonal has square 5² + 12² = 169.
3. Take the positive square root because this is a length.

Answer: 13 cm