Wentworth's Plane and Solid Geometry · Book VII, Props. I–III · Book 8: Book VII: Polyhedrons, Cylinders, and Cones
94. Sections of a prism, lateral area, and equal prisms
Goal: Your student can prove that parallel sections of a prism are equal, find its lateral area, and state when two prisms are equal.
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 I–III (§§ 605–611) with corollaries, and Exercises 630–633.
The lesson, step by step
Warm up
Ask: What is a right section of a prism? A truncated prism?
The new idea
The sections of a prism made by parallel planes cutting all the lateral edges are equal polygons.
The lateral area of a prism is equal to the product of a lateral edge by the perimeter of a right section.
Two prisms are equal if three faces including a trihedral angle of the one are respectively equal to three faces including a trihedral angle of the other, and are similarly placed.
Draw the figure
Draw a pentagonal prism with two slanting parallel sections. Then draw a right section, perpendicular to the edges.
Walk through the proof
I: the sides of the sections are parallel (§ 528) and, between parallels, equal (§ 180); their angles are equal (§ 534). So the polygons are equal.
II: each lateral face is a parallelogram whose base is a lateral edge E and whose altitude is a side of the right section. Add them: S = E × P.
III: the three face angles at A equal those at A', so the trihedral angles are equal (§ 582). Lay one on the other; the parallel edges then fall together and the prisms coincide.
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. 633. Find the lateral area of a right prism, if its altitude is 18 inches and the perimeter of its base 20 inches.
Then prove Ex. 631: the lateral faces of a right prism are rectangles.
Wrap up
Ask: How much paper does it take to wrap the sides (not the top and bottom) of a box 10 by 6 inches and 8 inches high? (Perimeter 32 × 8 = 256 sq in.)
Answers to the book's problems
Ex. 630: two opposite edges and the diagonals joining their ends form a parallelogram, whose diagonals bisect each other; every pair of diagonals shares a middle point. Ex. 631: the lateral edges are perpendicular to the base edges. Ex. 632: the section has its sides on two lateral faces parallel (§ 528) and the other two parallel to the edges. Ex. 633: 18 × 20 = 360 sq in.
Worked out for this site from the scan. If a number in your copy differs, trust the book.