Wentworth's Plane and Solid Geometry · Book VII, Props. XXIV–XXV · Book 8: Book VII: Polyhedrons, Cylinders, and Cones

105. Euler's theorem; the sum of the face angles

Goal: Your student can state and check Euler's theorem E + 2 = V + F, and find the sum of the face angles of a polyhedron.

⏱ About 40 minutes · 7 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)
  • Two or three stiff cards (to stand for planes) and a few pencils or straws (to stand for lines)

What's in the book

Propositions XXIV (Euler's) and XXV (§§ 662–663), General Theorems of Polyhedrons.

The lesson, step by step

  1. Warm up

    Ask: Count the faces, edges and corners of a box. (6, 12, 8.)

  2. The new idea

    In any polyhedron the number of edges increased by two is equal to the number of vertices increased by the number of faces.

    The sum of the face angles of any polyhedron is equal to four right angles taken as many times, less two, as the polyhedron has vertices.

  3. Draw the figure

    Make a table on paper: solid, F, V, E. Fill it in for a cube (6, 8, 12), a triangular prism (5, 6, 9), a square pyramid (5, 5, 8) and a tetrahedron (4, 4, 6). Check E + 2 = V + F for each.

  4. Walk through the proof

    XXIV: build the surface face by face. One face: E = V. Each new face added brings one more edge than vertices, until the last face, which adds neither. So E = V + F − 2.

    XXV: count the angles of all the faces as separate polygons, with an exterior angle at each corner: 2E corners give E × 4 right angles, less F × 4 right angles of exterior angles. So S = (E − F) × 4 rt. angles = (V − 2) × 4 rt. angles.

  5. Your student proves it back

    Close the book. Your student sketches the figure from memory (a simple perspective drawing is fine), letters it the same way, and gives the proof aloud: what is given, what is to be proved, then each step with its reason ("by § 501", "by § 507"). Point to each line and plane as it is named.

    If a step is missing, ask "How do we know that?" rather than supplying it.

  6. Exercises

    Check Prop. XXV on a cube: 8 vertices, so (8 − 2) × 360° = 2160°. Count directly: 6 faces × 360° = 2160°.

  7. Wrap up

    Ask: A solid has 12 faces and 30 edges. How many vertices? (20.)

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