Wentworth's Plane and Solid Geometry · Book VII, Props. IV–VI · Book 8: Book VII: Polyhedrons, Cylinders, and Cones
95. Oblique prisms and parallelopipeds
Goal: Your student can prove that an oblique prism equals a right prism on its right section, that opposite faces of a parallelopiped are equal and parallel, and that a diagonal plane halves it.
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 IV–VI (§§ 612–615) and Exercises 634–636.
The lesson, step by step
Warm up
Ask: What is the lateral area of a prism? (Lateral edge × perimeter of right section.)
The new idea
An oblique prism is equivalent to a right prism whose base is equal to a right section of the oblique prism, and whose altitude is equal to a lateral edge of the oblique prism.
The opposite lateral faces of a parallelopiped are equal and parallel.
The plane passed through two diagonally opposite edges of a parallelopiped divides the parallelopiped into two equivalent triangular prisms.
Draw the figure
Stack a deck of cards and push it slantwise: it leans like an oblique prism, but its volume has not changed. For Prop. IV, draw an oblique prism AD' with a right section FI, and a right prism of the same edge standing on FI.
Walk through the proof
IV: the two truncated prisms cut off at the bottom of each solid are equal (§ 610), since they have equal edges and the same right section. Each solid is one truncated prism plus the common part FD'.
V: BC is equal and parallel to AD, BF to AE, so the angles FBC and EAD are equal and their planes parallel (§ 534); the faces are equal parallelograms.
VI: the two triangular prisms are each equivalent to a right prism on half the right section (§ 612), and those right prisms are equal.
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. 634. Find the lateral area of a right prism, if its altitude is 20 inches, and its base is a triangle whose sides are 7 inches, 8 inches, and 9 inches, respectively.
Ex. 636. Find the total area of a right prism, if its altitude is 32 inches, and its base is a triangle whose sides are 12 inches, 14 inches, and 16 inches, respectively.
For the base of Ex. 636, use the formula for the area of a triangle from its three sides (Book IV).
Wrap up
Ask: If you slice a brick from one top edge to the opposite bottom edge, what are the two pieces? (Two equal triangular prisms.)
Answers to the book's problems
Ex. 634: 20 × 24 = 480 sq in. Ex. 635: 20 × 31 = 620 sq in. Ex. 636: lateral area 32 × 42 = 1344 sq in; each base √(21 · 9 · 7 · 5) = √6615, about 81.33 sq in; total about 1506.7 sq in.
Worked out for this site from the scan. If a number in your copy differs, trust the book.