Understand the cuboid

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.

Properties that make the calculations work

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.

Where the model is useful

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.

Measure the right thing

Internal capacity and external dimensions differ for a box with thick walls. Choose one consistent measurement convention.

Sketch the figure and mark every known measurement before choosing a calculator. A length measured along an edge is not automatically a perpendicular height. If a formula needs a radius, check that the value has not been measured across a full diameter. Write the unit beside each known quantity before entering it.

Scale and units

Multiplying every length by k multiplies any area by k² and any volume by k³. This is a useful check: doubling all dimensions cannot merely double an area. Inputs expressed in centimetres produce cm² for area and cm³ for volume; the unit selector relabels rather than converts entered numbers.

Continue learning

Wolfram MathWorld: Cuboid provides reference reading. Our summaries and examples are written for this portal and are not a specialist-reviewed textbook. Try a worked example or suggest a correction.