Wentworth's Plane and Solid Geometry · Book VII, Props. VII–IX · Book 8: Book VII: Polyhedrons, Cylinders, and Cones
96. Comparing rectangular parallelopipeds
Goal: Your student can prove that rectangular parallelopipeds are to each other as the products of their three dimensions.
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 VII–IX (§§ 616–621), the definition of dimensions, and Exercises 637–638.
The lesson, step by step
Warm up
Ask: How did Book IV compare two rectangles? (As the products of their bases and altitudes.)
The new idea
Two rectangular parallelopipeds having equal bases are to each other as their altitudes.
Two rectangular parallelopipeds having equal altitudes are to each other as their bases.
Two rectangular parallelopipeds are to each other as the products of their three dimensions.
Draw the figure
Build two stacks of sugar cubes or blocks on the same base, one 2 high and one 5 high. Count the cubes: the ratio is 2 : 5, the ratio of the heights.
Walk through the proof
VII: if the altitudes have a common measure, planes through the points of division cut the solids into m and n equal pieces. The incommensurable case uses limits, as in § 549.
VIII and IX: bring in a third solid Q that shares two dimensions with each. Then P : Q and Q : P' are known by Prop. VII; multiply the two ratios.
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. 637. Find the ratio of two rectangular parallelopipeds, if their altitudes are each 6 inches, and their bases 5 inches by 4 inches, and 10 inches by 8 inches, respectively.
Ex. 638. Find the ratio of two rectangular parallelopipeds, if their dimensions are 3, 4, 5, and 9, 8, 10, respectively.
Wrap up
Ask: If every edge of a box is doubled, how many times as large is the box? (8.)
Answers to the book's problems
Ex. 637: 120 : 480 = 1 : 4. Ex. 638: 60 : 720 = 1 : 12.
Worked out for this site from the scan. If a number in your copy differs, trust the book.