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

70. Area of a regular polygon and of a circle

Goal: Your student can prove that the area of a regular polygon is half the apothem times the perimeter, the area of a circle is πR², and related results on sectors and segments.

⏱ About 40 minutes · 7 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 X–XII (§§ 459–466) with corollaries.

The lesson, step by step

  1. Warm up

    Ask: What is π? What is the circumference of a circle of radius R?

  2. The new idea

    The area of a regular polygon is equal to half the product of its apothem by its perimeter.

    The area of a circle is equal to half the product of its radius by its circumference.

    The areas of two similar segments are to each other as the squares of their radii.

  3. Draw the figure

    Draw a regular hexagon in a circle and join the centre to every vertex, making six triangles. Mark the apothem as the altitude of one triangle.

  4. Walk through the proof

    X: the polygon is cut into triangles with the apothem as their common altitude; add their areas.

    XI: circumscribe a regular polygon; its area is half its apothem (the radius) times its perimeter. As the sides increase, the perimeter approaches the circumference and the area approaches the circle (§ 284).

    Corollary § 463: half of R × 2πR is πR².

    XII: similar sectors and similar triangles are both as the squares of the radii, so their differences are too.

  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. Exercises

    Compute: the area of a circle of radius 10 cm (about 314 sq cm), and the area of a sector of 90° in the same circle.

  7. Wrap up

    Ask: If the radius doubles, what happens to the area? (It becomes four times as large.)

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