Wentworth's Plane and Solid Geometry · Book VII, §§ 664–672 (Props. XXVI–XXVIII) · Book 8: Book VII: Polyhedrons, Cylinders, and Cones
106. Similar polyhedrons
Goal: Your student can define similar polyhedrons and prove that their volumes are as the cubes of homologous edges.
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
Definitions (§§ 664–665), Propositions XXVI–XXVIII (§§ 666–672) with corollaries, and Exercises 676–677.
The lesson, step by step
Warm up
Ask: How do the areas of similar polygons compare? (As the squares of homologous sides.)
The new idea
The volumes of two similar tetrahedrons are to each other as the cubes of their homologous edges.
The volumes of two similar polyhedrons are to each other as the cubes of any two homologous edges.
§§ 667–670: homologous edges and lines are proportional; faces and whole surfaces are as the squares of homologous edges.
Draw the figure
Build a cube from 1 block and another from 8 blocks (2 by 2 by 2). The edge doubled; count how the surface (squares showing) and volume (blocks) changed.
Walk through the proof
XXVI: cut both solids into tetrahedrons from homologous vertices; corresponding tetrahedrons are similar.
XXVII: by Prop. XXII the volumes are as SA × SB × SC to S'A' × S'B' × S'C', and each pair of edges has the same ratio, so the product is the cube of one ratio.
XXVIII: add up the tetrahedrons; each pair has the same ratio (§ 335).
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. 676. The homologous edges of two similar tetrahedrons are as 6 : 7. Find the ratio of their surfaces and of their volumes.
Ex. 677. If the edge of a tetrahedron is a, find the homologous edge of a similar tetrahedron twice as large.
Wrap up
Ask: A model ship is built at 1/100 of full size. How do its sails and its weight compare with the real ship? (1/10,000 of the area, 1/1,000,000 of the volume.)
Answers to the book's problems
Ex. 676: surfaces 36 : 49, volumes 216 : 343. Ex. 677: a∛2, about 1.26a.
Worked out for this site from the scan. If a number in your copy differs, trust the book.