Mathematics Year 7: Regular and irregular polyhedra: models and nets
Mathematics, Year 7 (Portugal): Two tests tell a regular polyhedron from an irregular one; then go from a net to the solid and back.
Regular or irregular: two tests at once
A polyhedron is regular when it passes two tests: all the faces are the same regular polygon, and the same number of faces meet at every vertex. The cube passes: 6 equal squares, 3 at each vertex. A pentagonal prism fails the first test: it has pentagons and rectangles. There are five regular polyhedra: tetrahedron, cube, octahedron, dodecahedron and icosahedron. All the others are irregular.
Why the regular ones are special
In a regular polyhedron every face and every corner looks the same: turn it and nothing tells you which face was on top. That is why dice are shaped like them, so that no face has an advantage. In an irregular polyhedron, faces or corners differ, and the differences are exactly what you have to explain. Among the five, only the cube is a prism, and the tetrahedron is the triangular pyramid whose faces are all equilateral triangles.
From the net to the solid: 8 triangles
A net has 8 equal equilateral triangles and folds into a regular polyhedron. Which one? Step 1: 8 pieces, so 8 faces. Step 2: the regular ones have 4, 6, 8, 12 and 20 faces, so it must be the octahedron. Step 3: 8 × 3 = 24 sides, glued in pairs: 24 ÷ 2 = 12 edges. Step 4: 8 × 3 = 24 triangle corners, 4 at each vertex: 24 ÷ 4 = 6 vertices. Check on the model: 6 vertices, 12 edges.
All faces equal is not enough
Glue two triangular pyramids with equilateral faces together along a base. You get a solid with 6 faces, all equal equilateral triangles. It looks regular, but it is not: at 2 vertices 3 faces meet, and at the other 3 vertices 4 faces meet. It fails the second test. It is a triangular bipyramid, with 5 vertices, 9 edges and 6 faces. Always run both tests: the faces and the corners.
Dice, models and nets
Dice with 4, 6, 8, 12 and 20 faces have the shapes of the five regular polyhedra. To build them use snap-together polygons, or draw a net: 4 triangles for the tetrahedron, 6 squares for the cube, 8 triangles for the octahedron, 12 pentagons for the dodecahedron, 20 triangles for the icosahedron. Before gluing, try to picture the fold: how many pieces will meet at each corner?
Test what you learned
Which sentence describes a regular polyhedron?
- a) Its two bases are regular polygons.
- b) All faces are the same regular polygon, and the same number of faces meet at every vertex.
- c) All its edges have the same length.
- d) All its faces are regular polygons, not necessarily equal.
Correct answer: b) All faces are the same regular polygon, and the same number of faces meet at every vertex. — Yes, both tests together. The cube (6 squares, 3 at each vertex) and the icosahedron (20 triangles, 5 at each vertex) pass them.
A net of 12 equal regular pentagons folds into a regular polyhedron, with 3 pentagons at each vertex. How many vertices does it have?
- a) 12
- b) 60
- c) 20
- d) 36
Correct answer: c) 20 — Yes: 12 × 5 = 60 pentagon corners, and 3 of them meet at each vertex, so 60 ÷ 3 = 20 vertices (the dodecahedron).
Ana glues two triangular pyramids with equilateral faces along a base: 6 faces, all equal equilateral triangles. She says it is a regular polyhedron. Is she right?
- a) No: 3 faces meet at 2 of the vertices, but 4 faces meet at the other 3.
- b) Yes: all the faces are equal equilateral triangles, and that is enough.
- c) Yes: all 9 edges have the same length.
- d) No: a regular polyhedron cannot have triangular faces.
Correct answer: a) No: 3 faces meet at 2 of the vertices, but 4 faces meet at the other 3. — Right: the second test fails, because the corners are not all alike. The solid has 5 vertices, 9 edges and 6 faces.
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