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

68. Similar regular polygons, and the apothem as a limit

Goal: Your student can prove that regular polygons of the same number of sides are similar, that their perimeters are as their radii, and that the apothem approaches the radius.

⏱ About 40 minutes · 6 steps

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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)

What's in the book

Propositions IV–VI (§§ 445–450) with corollaries.

The lesson, step by step

  1. Warm up

    Ask: Define the apothem and the angle at the centre of a regular polygon.

  2. The new idea

    Two regular polygons of the same number of sides are similar.

    The perimeters of two regular polygons of the same number of sides are to each other as the radii of their circumscribed circles, and also as the radii of their inscribed circles.

    If the number of sides of a regular inscribed polygon is indefinitely increased, the apothem of the polygon approaches the radius of the circle as its limit.

  3. Draw the figure

    Draw a circle with an inscribed square, then an inscribed octagon (bisect the arcs). Draw the apothem of each and compare it with the radius.

  4. Walk through the proof

    IV: each angle is the same fraction of the angle sum; the sides are all equal, so they are proportional.

    V: the triangles made by a side and two radii are similar, so the sides are as the radii and as the apothems.

    VI: OA − OP is less than AP, half a side; and a side can be made as small as we like by doubling the number of sides.

  5. 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.

  6. Wrap up

    Ask: What happens to the inscribed polygon as the number of sides grows? (It hugs the circle more and more closely.)

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