Wentworth's Plane and Solid Geometry · Book VII, §§ 702–721 (Props. XXXVI–XXXVIII) · Book 8: Book VII: Polyhedrons, Cylinders, and Cones
111. Cones
Goal: Your student can define a conical surface, a cone, a cone of revolution, slant height and frustum, and prove the first theorems about sections of a cone.
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)
- A sheet of paper, tape and scissors
What's in the book
Definitions of cones (§§ 702–716) and Propositions XXXVI–XXXVIII (§§ 717–721).
The lesson, step by step
Warm up
Ask: How is a pyramid related to a prism? How do you think a cone is related to a cylinder?
The new idea
A conical surface is the surface generated by a moving straight line which constantly touches a fixed curve and passes through a fixed point not in the plane of the curve.
Every section of a cone made by a plane passing through its vertex is a triangle.
Every section of a circular cone made by a plane parallel to the base is a circle.
§ 706: a right circular cone is a cone of revolution, made by turning a right triangle about one leg; the hypotenuse is the slant height.
Draw the figure
Roll a sheet of paper into a cone and tape it. Cut a right triangle from card, tape one leg to a pencil and spin it to see a cone of revolution.
Draw a cone with its altitude, slant height, and a frustum cut off by a parallel plane.
Walk through the proof
XXXVI: the plane cuts the base in a straight line BD; SB and SD are elements, so they are straight and lie in the plane.
XXXVII: the parallel section cuts each plane through the axis in a line parallel to a radius, making similar triangles; so every radius of the section is the same fraction of the base radius.
XXXVIII: exactly as for the cylinder, the cone is the limit of inscribed and circumscribed pyramids.
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.
Wrap up
Ask: What shape do you get if you slice an ice-cream cone straight across? Straight down through the tip?