Wentworth's Plane and Solid Geometry · Book VII, Props. XXII–XXIII · Book 8: Book VII: Polyhedrons, Cylinders, and Cones
104. Pyramids with a common trihedral angle; the truncated prism
Goal: Your student can prove that pyramids with an equal trihedral angle are as the products of the three edges, and find the volume of a truncated triangular prism.
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 XXII–XXIII (§§ 658–661) with corollaries.
The lesson, step by step
Warm up
Ask: State the frustum formula.
The new idea
The volumes of two triangular pyramids, having a trihedral angle of the one equal to a trihedral angle of the other, are to each other as the products of the three edges of these trihedral angles.
A truncated triangular prism is equivalent to the sum of three pyramids, whose common base is the base of the prism and whose vertices are the three vertices of the inclined section.
§ 660: the volume of a truncated right triangular prism is equal to the product of its base by one third the sum of its lateral edges.
Draw the figure
Cut a triangular prism of cheese, clay or a pencil eraser with a slanting cut: that is a truncated prism. Draw ABC-DEF with three lateral edges of different lengths.
Walk through the proof
XXII: place one pyramid in the other so the equal trihedral angles coincide. Take the faces S'AB and S'A'B' as bases; their ratio is SA × SB : S'A' × S'B' (§ 410), and the altitudes are as SC : S'C' (similar triangles).
XXIII: cut into E-ABC, E-ACD and E-CDF; slide the vertex of each along a line parallel to its base to turn them into D-ABC and F-ABC (§ 650).
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
Compute with § 660: a right triangular prism with base 10 sq in is cut so that its edges are 3, 4 and 8 in long. (10 × 5 = 50 cu in.)
Wrap up
Ask: What happens to § 660 if all three edges are equal? (It becomes base × height, an ordinary prism.)