Wentworth's Plane and Solid Geometry · Book VII, §§ 673–675 (Prop. XXIX) · Book 8: Book VII: Polyhedrons, Cylinders, and Cones

107. The five regular polyhedrons

Goal: Your student can prove that only five regular convex polyhedrons are possible and builds them from paper.

⏱ About 45 minutes · 6 steps

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You'll need

  • The book, open to these pages
  • A ruler (straightedge) and a pair of compasses
  • Sharp pencil, eraser and plain paper (or a geometry notebook)
  • Stiff paper or thin card, scissors, and tape or glue
  • A protractor

What's in the book

Regular polyhedrons (§ 673), Proposition XXIX (§ 674), and the paper patterns for the five solids (§ 675).

The lesson, step by step

  1. Warm up

    Ask: The face angles at a corner of a solid add to less than what? (360°.)

  2. The new idea

    A regular polyhedron is a polyhedron whose faces are equal regular polygons, and whose polyhedral angles are equal.

    Prop. XXIX: to determine the number of regular convex polyhedrons possible.

  3. Walk through the reasoning

    At each corner at least three faces meet, and the angles total less than 360°.

    Triangles (60°): 3, 4 or 5 can meet (180°, 240°, 300°); 6 would make 360°.

    Squares (90°): only 3.

    Pentagons (108°): only 3 (324°).

    Hexagons (120°): 3 already make 360°.

    So there are just five.

  4. Build them

    Follow § 675: copy the patterns onto stiff paper, cut on the full lines, fold on the dotted ones, and tape or glue the tabs. Start with the tetrahedron and the octahedron; the dodecahedron and icosahedron take longer and can be finished on another day.

  5. Check Euler

    For each finished solid, count F, V and E and check E + 2 = V + F.

  6. Wrap up

    Your student names the five solids and says how many faces of which shape each has.

Tip

Draw the patterns larger than in the book, with each edge at least 5 cm, so they are easy to fold.

Answers to the book's problems

Tetrahedron: 4 triangles, 4 vertices, 6 edges. Cube (hexahedron): 6 squares, 8, 12. Octahedron: 8 triangles, 6, 12. Dodecahedron: 12 pentagons, 20, 30. Icosahedron: 20 triangles, 12, 30.

Worked out for this site from the scan. If a number in your copy differs, trust the book.

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