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

101. The volume of a pyramid

Goal: Your student can prove that pyramids with equivalent bases and equal altitudes are equivalent and that the volume of any pyramid is one third base times altitude.

⏱ 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 XVII–XIX (§§ 650–654).

The lesson, step by step

  1. Warm up

    Ask: What is the volume of a prism?

  2. The new idea

    Two triangular pyramids having equivalent bases and equal altitudes are equivalent.

    The volume of a triangular pyramid is equal to one third of the product of its base by its altitude.

    The volume of any pyramid is equal to one third the product of its base by its altitude.

  3. Draw the figure

    For Prop. XVIII, draw a triangular prism ABC-EDS on the base of the pyramid S-ABC, and show how two planes split the prism into three pyramids.

    If you have modeling clay, make a triangular prism and cut it into the three pyramids.

  4. Walk through the proof

    XVII: inscribe prisms of the same thickness in both pyramids; corresponding sections are equivalent (§ 648), so the sums of the prisms are always equal, and so are their limits.

    XVIII: the prism is cut into three pyramids; any two of them have equal bases and the same altitude (§ 650), so all three are equivalent. Each is one third of B × H.

    XIX: cut any pyramid into triangular pyramids by planes through one lateral edge.

  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.

    Ask especially: in Prop. XVIII, which two pyramids are compared, and what makes their bases equal?

  6. Exercises

    Fill a hollow cone or pyramid (paper) with rice and pour it into a prism or cylinder of the same base and height, if you have one. It takes three fillings.

  7. Wrap up

    Ask: A pyramid and a prism have the same base and height. How do their volumes compare? (The pyramid is one third.)

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