Look a little closer.

One edge controls a three-dimensional world. These connections help explain why the figure behaves as it does, beyond simply entering numbers into a formula.

01 / A useful relationship

A cube is one of the five Platonic solids.

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.

02 / A different perspective

Its dual polyhedron is a regular octahedron.

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.

03 / A connection to explore

Doubling its edge multiplies surface area by four and volume by eight.

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.

Try the idea with numbers

Open the from an edge 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: Cube at Wolfram MathWorld.