Wentworth's Plane and Solid Geometry · Book V, Exs. 478–502 · Book 6: Book V: Regular Polygons and Circles
76. Constructions with circles and regular polygons
Goal: Your student circumscribes regular polygons, constructs circles of given size, and builds regular polygons on a given side.
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
Page 243: construction exercises 478–502.
The lesson, step by step
Warm up
Ask: Inscribe a square and a hexagon in a circle.
Circumscribing
Ex. 479. To circumscribe a square about a given circle.
Use tangents at the ends of two perpendicular diameters.
Circles of given size
Ex. 486. To construct a circle equivalent to the sum of two given circles.
Circles are as the squares of their radii, so use a right triangle, as in Book IV, Prop. XI.
Angles
Ex. 493. To construct an angle of 18° ; of 36° ; of 9°.
Use the decagon.
On a given side
Your student constructs a regular hexagon and a square with a side of 4 cm (Ex. 496, 497), and tries the pentagon (Ex. 499).
Wrap up
Your student shows the most accurate figure and explains it.
Answers to the book's problems
Ex. 486: the radius is the hypotenuse of a right triangle whose legs are the two given radii. Ex. 488: radius R√3, the hypotenuse of a right triangle with legs R and R√2. Ex. 493: the central angle of the decagon is 36°; bisect for 18° and 9°. Ex. 494: the central angle of the pentedecagon is 24°; bisect for 12° and 6°. Ex. 497: the side of the hexagon equals the radius, so draw a circle of that radius and step it round.
Worked out for this site from the scan. If a number in your copy differs, trust the book.