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

103. The frustum of a pyramid

Goal: Your student can prove the formula V = ⅓H(B + b + √(B × b)) for the frustum of a pyramid.

⏱ 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)

What's in the book

Propositions XX–XXI (§§ 655–657).

The lesson, step by step

  1. Warm up

    Ask: What is a frustum? What is the mean proportional between 4 and 9? (6.)

  2. The new idea

    The frustum of a triangular pyramid is equivalent to the sum of three pyramids whose common altitude is the altitude of the frustum and whose bases are the lower base, the upper base, and the mean proportional between the two bases of the frustum.

    § 656: V = ⅓ H × (B + b + √(B × b)).

    Prop. XXI extends this to the frustum of any pyramid.

  3. Draw the figure

    Draw the triangular frustum ABC-DEF and the two planes AEC and EDC that cut it into three pyramids.

  4. Walk through the proof

    XX: E-ABC has base B and altitude H; C-EDF has base b and altitude H. The middle pyramid E-ADC is compared with each: E-ABC : E-ADC = AB : DE, and E-ADC : E-DFC = AC : DF; these ratios are equal (§ 645). So E-ADC is the mean proportional between the other two, and its volume is ⅓ H √(B × b).

    XXI: replace any frustum by a triangular one with equivalent bases and the same altitude (§ 648).

  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.

    The algebra is the hard part. It is enough that your student can explain why the middle pyramid is the mean proportional of the other two.

  6. Exercises

    Compute: a frustum with square bases 6 ft and 3 ft on a side and altitude 4 ft. (⅓ × 4 × (36 + 9 + 18) = 84 cu ft.)

  7. Wrap up

    Ask: What does the formula give if the upper base shrinks to a point? (⅓ H B, a pyramid.)

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