Wentworth's Plane and Solid Geometry · Book V, Exs. 452–477 · Book 6: Book V: Regular Polygons and Circles
75. Theorems on regular polygons and circles
Goal: Your student proves original theorems about regular polygons inscribed in and circumscribed about circles.
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
Pages 241–242: theorem exercises 452–477 on regular polygons, including formulas for the sides of the inscribed pentagon, octagon and dodecagon.
The lesson, step by step
Warm up
Ask: How are the side and the radius of an inscribed regular hexagon related? (They are equal.)
Together
Ex. 455. The side of a circumscribed equilateral triangle is equal to twice the side of the similar inscribed triangle.
Hint: the inscribed triangle's vertices are the midpoints of the sides of the circumscribed one.
On their own
Your student picks two or three: Ex. 456, 457, 460, 464 or 473.
Ex. 460. The area of an inscribed equilateral triangle is equal to half the area of the inscribed regular hexagon.
A formula
Check Ex. 464 or 465 by measuring: draw a regular octagon in a circle of radius 5 cm and measure a side.
Wrap up
Your student explains one proof aloud.
Answers to the book's problems
Ex. 455: the tangent points (vertices of the inscribed triangle) bisect the sides of the circumscribed one, so its sides join midpoints and are half as long (§ 189). Ex. 460: each side of the triangle cuts off a small triangle at a vertex of the hexagon; with the centre O, that small triangle and the triangle on the same side toward O make a rhombus (all four sides equal R), so they are equal. Hence the triangle equals the three pieces cut off, and is half the hexagon. Ex. 464: use Prop. XVIII with the square's side R√2. Ex. 465: use Prop. XVIII with the hexagon's side R.
Worked out for this site from the scan. If a number in your copy differs, trust the book.