Wentworth's Plane and Solid Geometry · Book VII, Props. XV–XVI · Book 8: Book VII: Polyhedrons, Cylinders, and Cones
100. Parallel sections of a pyramid; the pyramid as a limit
Goal: Your student can prove that a plane parallel to the base cuts the edges and altitude proportionally in a similar section, and explain why a pyramid is the limit of stacked prisms.
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 XV–XVI (§§ 645–649) with corollaries, and Exercises 660–661.
The lesson, step by step
Warm up
Ask: Lateral area of a regular pyramid?
The new idea
If a pyramid is cut by a plane parallel to the base:
1. The edges and altitude are divided proportionally.
2. The section is a polygon similar to the base.
§ 646: the section is to the base as the square of its distance from the vertex is to the square of the altitude.
Prop. XVI (§ 649): the volume of a triangular pyramid is the limit of the sum of a series of inscribed or circumscribed prisms of equal altitude.
Draw the figure
Build a "step pyramid" from blocks or stacked cards of shrinking size. The steps stand for the inscribed prisms of Prop. XVI; with thinner steps the stack hugs the pyramid more closely.
Walk through the proof
XV: the sides of the section are parallel to those of the base (§ 528), so the small and large triangles at the vertex are similar, giving the proportions; the polygons have proportional sides and equal angles (§ 534).
XVI: each inscribed prism equals the circumscribed prism just above it, so the two totals differ only by the bottom prism, which can be made as small as we like. The pyramid lies between them, so it is their common limit.
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.
Exercises
Ex. 661. The section of a pyramid made by a plane parallel to the base is half as large as the base. Find the ratio of the segments into which the altitude is divided by the plane.
Wrap up
Ask: If a plane cuts a pyramid halfway up, how does the section compare with the base? (One quarter as large.)
Answers to the book's problems
Ex. 660: the section is a quarter of the slant height from the vertex, so base : section = 16 : 1. Ex. 661: the distance from the vertex is the altitude ÷ √2, so the upper segment : lower segment = 1 : (√2 − 1), about 2.414 : 1.
Worked out for this site from the scan. If a number in your copy differs, trust the book.