Wentworth's Plane and Solid Geometry · Book V, §§ 429–444 (Props. I–III) · Book 6: Book V: Regular Polygons and Circles
67. Regular polygons and their circles
Goal: Your student can define a regular polygon and its radius, apothem and centre, and prove that circles can be inscribed in and circumscribed about every regular polygon.
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
The definition of a regular polygon (§ 429), Propositions I–III (§§ 430–444) and the definitions of radius, apothem, centre and angle at the centre.
The lesson, step by step
Warm up
Ask: What is the sum of the angles of a hexagon? Each angle of a regular hexagon?
The new idea
A regular polygon is a polygon which is both equilateral and equiangular.
An equilateral polygon inscribed in a circle is a regular polygon.
A circle may be circumscribed about, and a circle may be inscribed in, any regular polygon.
Prop. III (§ 439): if the circumference is divided into equal arcs, the chords form a regular inscribed polygon and the tangents at the points of division a regular circumscribed polygon.
Draw the figure
Draw a circle and step the radius around it six times with the compasses. Join the points: a regular hexagon. Draw tangents at the six points: a larger regular hexagon. Mark the radius and apothem.
Walk through the proof
I: equal chords cut off equal arcs; each angle of the polygon stands on an arc made of the same number of these, so the angles are equal.
II: pass a circle through three vertices; equal triangles show that it passes through the fourth, then the fifth, and so on. The sides are equal chords, so they are equally far from the centre, giving the inscribed circle.
III: equal arcs give equal chords; the tangents make equal isosceles triangles.
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.
Wrap up
Read §§ 436–444 (corollaries). Ask: What is the angle at the centre of a regular octagon? (45°.)