Understand the cube

A cube is a solid with six congruent square faces, twelve equal edges and eight vertices. Three mutually perpendicular edges meet at each vertex. One edge length determines its entire metric geometry.

Properties that make the calculations work

Its face diagonal is a√2 and its space diagonal is a√3. Total surface area counts all six faces. Volume counts the enclosed three-dimensional space and therefore uses cubic units.

Where the model is useful

Use an edge or an inverse measurement to size an ideal block. Packaging calculations may also need wall thickness and internal dimensions; a mathematical cube does not include either automatically.

Measure the right thing

A face diagonal and a space diagonal are different measurements. Surface area is not measured in cubic units.

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.

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Wolfram MathWorld: Cube 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.