Wentworth's Plane and Solid Geometry · Book VII, §§ 676–694 (Props. XXX–XXXI) · Book 8: Book VII: Polyhedrons, Cylinders, and Cones

108. Cylinders

Goal: Your student can define a cylindrical surface, a cylinder, a right circular cylinder and its parts, and prove that sections through an element are parallelograms and that the bases are equal.

⏱ About 40 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)
  • Two or three stiff cards (to stand for planes) and a few pencils or straws (to stand for lines)
  • A can or a cardboard tube

What's in the book

Definitions of cylinders (§§ 676–689), Propositions XXX–XXXI (§§ 690–694).

The lesson, step by step

  1. Warm up

    Ask: What is a prism? How is a can like a prism, and how is it different?

  2. The new idea

    A cylinder is a solid bounded by a cylindrical surface and two parallel plane surfaces.

    Every section of a cylinder made by a plane through an element is a parallelogram.

    The bases of a cylinder are equal.

    § 683: a right circular cylinder is called a cylinder of revolution, because it can be made by turning a rectangle about one side.

  3. Draw the figure

    Hold a pencil upright and move it around a coin on the table, keeping it upright: it sweeps out a cylindrical surface (§ 676). Spin a rectangle of card on a pencil to see a cylinder of revolution.

    Draw a cylinder with an inscribed and a circumscribed prism (§§ 687–688).

  4. Walk through the proof

    XXX: a line through B parallel to the element AD is also an element (§ 676), so it lies in both the plane and the surface; it is BC. AB and DC are parallel (§ 528). So ABCD is a parallelogram.

    XXXI: take any three points A, B, E of the lower base and the elements through them. The triangles ABE and DCG are equal (§ 150); lay one base on the other with these triangles together, and every point of one edge falls on the other.

  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. Wrap up

    Ask: Name three cylinders in your kitchen. Which are right circular cylinders?

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