Solid Geometry Notebook

Solid geometry is the part of mathematics that measures what has thickness: the solids you can hold, wrap, cut open, and stack. This notebook catalogues the classics — the five Platonic solids, the eleven nets of a cube, the formulas for volume and surface area, and the figures that tile a floor or fill a room — each entry drawn up like a sheet from a drafting class.

qubyn.comCallout: V − E + F = 2 holds for every convex polyhedron.Callout: exactly 11 of the 35 hexominoes fold into a cube.

Specification table

Data sheet: counts for the regular solids and one space-filling relative
Solid	Faces	Edges	Vertices	Shape of each face
Tetrahedron	4	6	4	Equilateral triangle
Cube (hexahedron)	6	12	8	Square
Octahedron	8	12	6	Equilateral triangle
Dodecahedron	12	30	20	Regular pentagon
Icosahedron	20	30	12	Equilateral triangle
Rhombic dodecahedron	12	24	14	Rhombus

Data sheet: counts for the regular solids and one space-filling relative

Figure gallery — the catalogue plate

Each card is lettered like a drawing: the tag gives the face count, the title names the solid, and the caption block carries the key facts.

Fig. 1 — The five regular solids as a dimensioned plate; callouts letter faces, edges, and vertices.

Fig. 2 — One cube net, the 1-4-1 pattern: four squares in a row, one attached above and one below.

Fig. 3 — The square-cube law: doubling edge length multiplies surface area by 4 and volume by 8.

What makes a solid 'regular'?

A Platonic solid is a convex polyhedron whose faces are all the same regular polygon, arranged the same way at every vertex.

Regularity is a strict dress code: every face a regular polygon, all faces identical, every vertex alike. A square pyramid fails the code — its base is square, its sides triangular, so its vertices are not all the same kind of corner.

Five solids pass: tetrahedron, cube, octahedron, dodecahedron, icosahedron. They pair off by duality — swap faces for vertices and the cube becomes the octahedron, the dodecahedron becomes the icosahedron, while the tetrahedron stays itself.

Every one of them balances the same ledger: vertices minus edges plus faces equals two. Euler's formula holds for all convex polyhedra, but the five regular ones are its cleanest illustrations.

Tetrahedron — 4 triangular faces, 3 at each vertex

Cube — 6 square faces, 3 at each vertex

Octahedron — 8 triangular faces, 4 at each vertex

Dodecahedron — 12 pentagonal faces, 3 at each vertex

Icosahedron — 20 triangular faces, 5 at each vertex

Dossier — the notebook in three parts

Why are there exactly five?

Because the face angles around a vertex must sum to less than 360°, and only five combinations manage that.

At least three faces meet at every vertex of a convex solid, and their angles must total less than a full turn — otherwise the corner lies flat or refuses to close. That one inequality is the entire census office of regular solids.

Run the count. Triangles at 60°: three, four, or five per vertex give 180°, 240°, 300° — the tetrahedron, octahedron, and icosahedron; six make 360° and lie flat. Squares: three give the cube, four lie flat. Pentagons at 108°: three give the dodecahedron, four overshoot. Hexagons fail immediately.

So the five are not a Greek aesthetic preference. They are the complete solution set of a single inequality, which is why no sixth has ever been found, in any century.

How does a flat sheet become a solid?

Cut a polyhedron along some edges and it opens into one connected piece — a net; folding runs the film backwards.

Every classic solid flattens this way. Whether absolutely every convex polyhedron has a non-overlapping edge-cut net remains an open question in full generality, but for the cube the answer is completely enumerated: eleven distinct nets.

The enumeration is a sieve. Six squares join edge-to-edge in 35 arrangements, the hexominoes; tested by folding, eleven succeed and twenty-four fail by overlapping a face or leaving one open.

Nets are working instruments. Cartons, sheet-metal blanks, and sewing patterns are nets with tabs, and the area of the flat sheet is exactly the surface area of the finished solid — which is why net problems and surface-area problems are the same problem twice.

A net is one connected piece — no detached faces
Uncut edges become fold lines; cut edges become the boundary
No cube net contains a 2×2 block of squares
No cube net holds five squares in a straight row

How do volume and surface area relate?

Surface area measures the wrapping; volume measures what the wrapping holds, and scaling drives the two apart.

For a cube of edge a the surface is 6a² and the volume a³. Double the edge and the surface quadruples while the volume octuples — area scales with the square of size, volume with the cube.

The other standards fit on one line each: a sphere carries surface 4πr² and volume (4/3)πr³; a cylinder, volume πr²h; any cone or pyramid, exactly one-third of base area times height — a result older than Euclid.

The two measures answer different questions. Surface area is a net problem — the size of the flat blank. Volume is a filling problem — the count of unit cubes inside. Confusing them is the classic error of the subject.

Short answers to standing questions

Why are there only five Platonic solids?

Because the angles meeting at each vertex must total less than 360°, and only five choices of regular polygon and count satisfy that. Three, four, or five triangles work; three squares; three pentagons — everything else lies flat or overshoots.

How many nets does a cube have?

Eleven. Six squares can be joined edge-to-edge in 35 distinct arrangements (the hexominoes), and exactly eleven of them fold into a cube without overlapping faces.

What is the difference between surface area and volume?

Surface area is the amount of wrapping; volume is the amount of room inside. When a solid doubles in size, its surface area grows fourfold while its volume grows eightfold — the square-cube law.

Which solids can fill space with no gaps?

Among the Platonic solids, only the cube. Copies of the rhombic dodecahedron also pack perfectly — the geometry at the capped end of honeycomb cells — and tetrahedra mixed with octahedra fill space in the octet-truss pattern.