A cube is a regular solid with six congruent square faces. It has twelve equal edges and eight vertices, with three edges meeting at each vertex. These counts describe its structure; the common edge length determines its area, volume and diagonals.
Faces, edges and vertices
A cube is one of the five Platonic solids. At each vertex, the three face angles are right angles; counting four corners on each of six faces gives 24 face-angle incidences. That count does not mean there are 24 vertices. Euler’s relation for this solid is vertices minus edges plus faces equals two.
One face, the lateral surface and the entire surface
Let a be the edge length. One face has square area. Choosing two opposite faces as bases leaves four lateral faces; the total surface includes all six. Volume multiplies length, width and height, each equal to a.
The last two formulas distinguish a face diagonal from a body diagonal. The first crosses one square face. The second joins opposite vertices through the cube’s center. Pythagoras gives the face diagonal, then combines it with another edge for the body diagonal. There are four body diagonals.
Worked example with edge 4 cm
Squaring 4 gives one face’s area, 16 square centimetres. Multiply by four or six for lateral or total surface. Cube 4 for the volume. Multiply 4 by the appropriate square root for each diagonal.
The lateral area and volume both have numerical value 64 here, but their units and meanings differ. One is in square centimetres and the other in cubic centimetres.
Scaling an edge changes area and volume differently
Doubling every edge multiplies all lengths by two, every surface area by four and volume by eight. Do not multiply volume by only two. These formulas assume an ideal solid or internal dimensions when describing capacity; an open box lacks its missing face. For more surface detail, see surface area of a cube.
