Wentworth's Plane and Solid Geometry · Book VII, Props. X–XIII · Book 8: Book VII: Polyhedrons, Cylinders, and Cones
97. Volumes of parallelopipeds and prisms
Goal: Your student can prove that the volume of a rectangular parallelopiped is the product of its dimensions, and that the volume of any prism is base times altitude.
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 X–XIII (§§ 622–629) with corollaries and the scholium, and Exercises 639–642.
The lesson, step by step
Warm up
Ask: Two rectangular parallelopipeds are to each other as ... ?
The new idea
The volume of a rectangular parallelopiped is equal to the product of its three dimensions.
The volume of any parallelopiped is equal to the product of its base by its altitude.
The volume of any prism is equal to the product of its base by its altitude.
Draw the figure
Build a 3 by 5 by 2 block of cubes (or draw one with the cubes marked), as in the scholium § 625.
Walk through the proof
X: compare P with the unit cube U by Prop. IX: P : U = abc : 1.
XI: change the oblique parallelopiped step by step into a right one and then a rectangular one, keeping base area and altitude (§ 612 twice).
XII: a triangular prism is half of a parallelopiped on the same edges (§ 615).
XIII: planes through one lateral edge cut any prism into triangular prisms of the same altitude; add their volumes.
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
Prove Ex. 640 together:
Ex. 640. The square of a diagonal of a rectangular parallelopiped is equal to the sum of the squares of the three dimensions.
Hint: use the right triangle twice, first on the base, then standing up.
Wrap up
Ask: A fish tank is 30 by 12 inches and 15 inches deep. How many cubic inches of water? (5400.)
Answers to the book's problems
Ex. 640: the diagonal of the base is √(a² + b²); with the height c it makes a right triangle whose hypotenuse is the diagonal, so d² = a² + b² + c². Ex. 642: the prism is half of a parallelopiped having that face as base and the perpendicular as altitude.
Worked out for this site from the scan. If a number in your copy differs, trust the book.