Wentworth's Plane and Solid Geometry · Book V, Props. XIII–XVI · Book 6: Book V: Regular Polygons and Circles
71. Inscribing the square, hexagon, decagon and pentedecagon
Goal: Your student can inscribe a regular square, hexagon, decagon and pentedecagon in a circle and prove each construction.
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 protractor
What's in the book
Propositions XIII–XVI (§§ 467–476), with corollaries, and Exercises 438–443.
The lesson, step by step
Warm up
Ask: Divide a line in extreme and mean ratio (Book III, Prop. XL).
The new idea
To inscribe a square in a given circle.
To inscribe a regular hexagon in a given circle.
To inscribe a regular decagon in a given circle.
Prop. XVI (§ 475): to inscribe a regular polygon of fifteen sides.
Draw the figure
In three circles: draw two perpendicular diameters (square); step the radius six times round (hexagon); and divide a radius OC in extreme and mean ratio at S, then step OS around the circle (decagon). Join alternate vertices of the decagon for a pentagon.
Walk through the proof
XIII: the four angles are in semicircles, so right angles, and equal arcs give equal sides.
XIV: the triangle made by two radii and a chord equal to the radius is equilateral, so the central angle is 60°, one sixth of the circle.
XV: triangles OCB and BCS are similar, which makes the angle at O one fifth of two right angles, 36°, one tenth of the circle.
XVI: an arc of one sixth minus an arc of one tenth is one fifteenth.
Your student proves it back
Close the book. Your student redraws the figure from memory, letters it the same way, and does each construction and explains why it works: 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.
Exercises
Ex. 438. The area of a circumscribed square is equal to twice the area of the inscribed square.
Then check the formulas of Ex. 440–442 for the triangle, square and hexagon by measuring your drawings with R = 5 cm.
Wrap up
Ask: How would you inscribe a regular polygon of 24 sides? (Hexagon, then bisect the arcs twice.)
Answers to the book's problems
Ex. 438: the inscribed square has the diameter 2R for its diagonal, so its area is 2R² (half the product of the diagonals); the circumscribed square has side 2R and area 4R².
Worked out for this site from the scan. If a number in your copy differs, trust the book.