Mathematics Year 7: Why there are only five regular polyhedra
Mathematics, Year 7 (Portugal): Add up the angles at a corner and see, polygon by polygon, which regular polyhedra can exist.
A corner has to close
At a vertex of a polyhedron at least 3 faces meet, and the angles of those faces around the vertex must add up to less than 360°. If they add up to exactly 360°, the pieces lie flat, like floor tiles. If they add up to more, they do not fit. The angles of the regular polygons are: triangle 60°, square 90°, pentagon 108°, hexagon 120°. With these numbers you decide which regular polyhedra can exist.
Why 360° is the border
Picture the polygons around a point on a table. With 360° they cover everything around it and stay flat. To lift a corner, some of that angle has to be missing: the missing bit lets the faces lean towards each other. The smaller the sum, the sharper the corner. So “how many regular polyhedra exist?” becomes an angle sum, done polygon by polygon. The sum shows there cannot be more; that the five exist, you see by building them.
Testing each polygon
Triangle (60°): 3 faces give 180°, 4 give 240°, 5 give 300°, so they work (tetrahedron, octahedron, icosahedron); 6 give 360°, flat. Square (90°): 3 give 270°, it works (cube); 4 give 360°, flat. Pentagon (108°): 3 give 324°, it works (dodecahedron); 4 give 432°, too much. Hexagon (120°): 3 already give 360°, flat. With 7 or more sides, 3 faces already pass 360°. Total: 3 + 1 + 1 = 5.
360° “fills” the corner
Put 6 equilateral triangles around a point: 6 × 60° = 360°. “It fits perfectly, so it must make a solid!” But it makes a flat hexagon: all the triangles stay in one plane and there is no corner. The same happens with 4 squares (4 × 90° = 360°) or 3 hexagons (3 × 120° = 360°). That is how you tile a floor, not how you build a solid. The corner of a polyhedron needs less than 360°.
The football
The classic black-and-white football is a truncated icosahedron, one of the Archimedean solids: 12 regular pentagons and 20 regular hexagons, with 60 vertices, 90 edges and 32 faces. It is not regular, because it has two kinds of faces, but at every vertex 1 pentagon and 2 hexagons meet: 108° + 120° + 120° = 348°, less than 360°, so the corner closes. The same angle sum works for other solids.
Test what you learned
How many regular polyhedra exist, and what explains it?
- a) Infinitely many: there is one for every regular polygon.
- b) Three: the tetrahedron, the cube and the octahedron.
- c) Four: one for each kind of face, triangle, square, pentagon and hexagon.
- d) Five: the angles at a vertex must add up to less than 360°, and only five combinations do.
Correct answer: d) Five: the angles at a vertex must add up to less than 360°, and only five combinations do. — Yes: 3, 4 or 5 triangles, 3 squares and 3 pentagons work, and no other combination stays below 360°.
Which group of faces can meet at a vertex of a regular polyhedron? (Angles: square 90°, pentagon 108°, hexagon 120°.)
- a) 3 regular hexagons
- b) 3 regular pentagons
- c) 4 squares
- d) 4 regular pentagons
Correct answer: b) 3 regular pentagons — 3 × 108° = 324°, which is less than 360°. The corner closes, and this is the corner of the dodecahedron.
Rui puts 6 equilateral triangles around a point: 6 × 60° = 360°. What happens when he tries to close the corner?
- a) It makes the corner of the icosahedron.
- b) It makes a hexagonal pyramid with equilateral faces.
- c) They lie flat and make a hexagon: there is no corner.
- d) They overlap, because 360° is too much.
Correct answer: c) They lie flat and make a hexagon: there is no corner. — Right: 360° is a full turn, so nothing is left to lift the faces. A corner of a polyhedron needs less than 360°.
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