Wentworth's Plane and Solid Geometry · Book V, Props. VII–IX · Book 6: Book V: Regular Polygons and Circles
69. The circumference as a limit, and π
Goal: Your student can explain why the circumference is the limit of inscribed and circumscribed perimeters, prove that circumferences are as their radii, and define π.
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 tin can, a string and a tape measure
What's in the book
Propositions VII–IX (§§ 451–458): the arc as the shortest enveloping line, circumference and area as limits, and the ratio of circumference to diameter.
The lesson, step by step
Warm up
Ask: State the test for a limit (Book II, § 275).
The new idea
An arc of a circle is less than any line which envelops it and has the same extremities.
Two circumferences have the same ratio as their radii.
Prop. VIII (§ 454): the circumference is the limit of the perimeters of regular inscribed and circumscribed polygons, and the area of the circle is the limit of their areas.
Draw the figure
Draw a circle with an inscribed and a circumscribed square. Bisect to make octagons. Show how the inside perimeter grows and the outside one shrinks, squeezing the circumference between them.
Walk through the proof
VII: among enveloping lines there must be a shortest; any line other than the arc can be shortened by a tangent.
VIII: the difference of the two perimeters, (P/R)(R' − R), can be made as small as we like, and the circumference is always between them.
IX: the perimeters of similar inscribed polygons are always as the radii; their limits are the circumferences (§ 285).
Corollary § 456: the ratio of circumference to diameter is the same for every circle; it is called π, and C = 2πR.
Your student proves it back
Close the book. Your student redraws the figure from memory, letters it the same way, and gives the proof aloud: what is given, what is to be proved, then each step with its reason ("because the radii are equal", "by § 143"). Point to each line or angle as it is named.
If a step is missing, ask "How do we know that?" rather than supplying it.
The limit arguments are the hard part. It is enough that your student can explain the squeeze between inner and outer polygons.
Exercises
Wrap a string around a can, measure it, and divide by the diameter. You should get a little over 3.
Wrap up
Ask: If one circle has twice the radius of another, what about its circumference? (Twice.)