Look a little closer.

Length, width and height in harmony. These connections help explain why the figure behaves as it does, beyond simply entering numbers into a formula.

01 / A useful relationship

Its twelve edges come in three groups of four equal lengths.

Opposite faces are congruent. The space diagonal connects opposite vertices through the interior and is √(a²+b²+c²). Total surface area includes each of the three distinct face areas twice.

02 / A different perspective

The space diagonal follows two consecutive uses of the Pythagorean theorem.

A cuboid, or rectangular box, is a solid with six rectangular faces and three perpendicular edge directions. Its length, width and height are independent positive measurements. A cube is the equal-edge special case.

03 / A connection to explore

For a fixed volume among cuboids, the cube minimises total surface area.

Use the three-edge tool for an ideal box. Inverse tools find a missing edge from volume, surface area or space diagonal together with two known edges. A volume alone cannot determine all dimensions.

Try the idea with numbers

Open the three perpendicular edges calculator, note its outputs, then multiply each length input by two. Volume should increase by eight and surface area by four, when every relevant length is doubled. Angle inputs and dimensionless ratios remain unchanged. This scaling experiment checks the structure of the formula rather than the precision of any real measurement.

Further reading: Cuboid at Wolfram MathWorld.