Wentworth's Plane and Solid Geometry · Book V, Props. XVII–XIX · Book 6: Book V: Regular Polygons and Circles

72. Similar inscribed polygons, doubling the sides, and computing π

Goal: Your student can inscribe a polygon similar to a given regular polygon, find the side of the polygon with twice as many sides, and follow the computation of π.

⏱ About 40 minutes · 7 steps

Open it at the Internet Archive ↗

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)

What's in the book

Propositions XVII–XIX (§§ 477–481) and Exercise 444. Prop. XIX computes π from polygons of 6 up to 768 sides.

The lesson, step by step

  1. Warm up

    Ask: Inscribe a regular hexagon and explain why it works.

  2. The new idea

    To inscribe in a given circle a regular polygon similar to a given regular polygon.

    Given the side and the radius of a regular inscribed polygon, to find the side of the regular inscribed polygon of double the number of sides.

    To find the numerical value of the ratio of the circumference of a circle to its diameter.

  3. Draw the figure

    For Prop. XVIII: draw a circle with centre O, a chord AB, and the diameter DH perpendicular to AB through its midpoint C. Join AD and AH.

  4. Walk through the proof

    XVII: copy the angle at the centre of the given polygon; equal central angles give the same number of sides (§ 445).

    XVIII: OC = √(R² − ¼a²), so DC = R − OC; in the right triangle DAH, AD² = DH × DC. With R = 1 this gives the formula in § 479.

    XIX: start with the hexagon (side 1 when R = 1) and apply the formula again and again. The perimeters approach 2π.

  5. Your student proves it back

    Your student explains in words how π was found: start with a hexagon, keep doubling the number of sides, watch the perimeter settle down.

  6. Exercises

    With a calculator, repeat the book's table: start with s = 1 and use s(new) = √(2 − √(4 − s²)). After seven steps, multiply by the number of sides and divide by 2.

  7. Wrap up

    Ask: Why do we use 3.1416 for π, but never an exact fraction? (Read § 481: π is incommensurable.)

@teddyrookbook GitHub