Wentworth's Plane and Solid Geometry · Book VII, Props. XXXIX–XLI · Book 8: Book VII: Polyhedrons, Cylinders, and Cones
112. Area and volume of a cone
Goal: Your student can prove S = πRL and V = ⅓πR²H for a cone of revolution, and compare similar cones.
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)
- A paper cone and scissors
What's in the book
Propositions XXXIX–XLI (§§ 722–726) with corollaries.
The lesson, step by step
Warm up
Ask: Lateral area and volume of a regular pyramid?
The new idea
The lateral area of a cone of revolution is equal to half the product of the slant height by the circumference of the base.
The volume of a circular cone is equal to one third the product of its base by its altitude.
§ 723 and § 725: S = πRL, T = πR(L + R), V = ⅓πR²H.
Draw the figure
Cut a paper cone along a slant height and flatten it: it is a sector of a circle of radius L, with arc equal to the circumference of the base. Its area is ½ × arc × radius, the same as Prop. XXXIX.
Walk through the proof
XXXIX: for a circumscribed regular pyramid S' = ½ P × L always; as the sides increase, S' approaches S and P approaches C (§ 720).
XL: for an inscribed pyramid V' = ⅓ B' × H; take the limit.
XLI: in similar cones H : H' = R : R' = L : L'; put these into the formulas.
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 the lateral area and volume of a cone with radius 3 and altitude 4. (L = 5; S = 15π ≈ 47.1; V = 12π ≈ 37.7.)
Wrap up
Ask: A cone and a cylinder have the same base and height. Compare their volumes. (The cone is one third.)