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

109. Area and volume of a cylinder

Goal: Your student can prove that the cylinder is the limit of inscribed prisms and derive S = 2πRH, T = 2πR(H + R) and V = πR²H.

⏱ 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)
  • A calculator (optional)
  • A can with a paper label, a ruler

What's in the book

Propositions XXXII–XXXV (§§ 695–701) with corollaries.

The lesson, step by step

  1. Warm up

    Ask: Area and circumference of a circle of radius R?

  2. The new idea

    The lateral area of a circular cylinder is equal to the product of the perimeter of a right section of the cylinder by an element.

    The volume of a circular cylinder is equal to the product of its base by its altitude.

    § 698 and § 700: for a cylinder of revolution, S = 2πR × H, T = 2πR(H + R), V = πR²H.

    Prop. XXXV: similar cylinders of revolution have areas as the squares, and volumes as the cubes, of their altitudes or radii.

  3. Draw the figure

    Peel the label off a can and lay it flat: it is a rectangle, one side the circumference, the other the height. That is § 697.

  4. Walk through the proof

    XXXII: by doubling the sides of the base polygon, the prism can be made to fit the cylinder as closely as we like (§ 454), so the volume, lateral area and right-section perimeter of the prism approach those of the cylinder.

    XXXIII and XXXIV: for the inscribed prism, S' = P' × E and V' = B' × H always; pass to the limit (§ 284).

    XXXV: in similar cylinders H : H' = R : R'; put this into the formulas.

  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

    Measure a can: its diameter and height. Compute its label area and its volume. Check the volume against what the label says (1 cu in is about 16.4 ml).

  7. Wrap up

    Ask: If a can is twice as tall and twice as wide, how much more does it hold? (8 times.)

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