Wentworth's Plane and Solid Geometry · Book I, §§ 191–207 (Props. XXXIX–XL) · Book 2: Book I: Rectilinear Figures
17. Polygons and the sum of their angles
Goal: Your student knows the polygon vocabulary and can prove the formulas for the interior and exterior angles of a 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
Definitions for polygons in general (§§ 191–204) and Propositions XXXIX and XL (§§ 205–207), with Exercises 17 and 18.
The lesson, step by step
Warm up
Ask: What do the angles of a triangle add up to? Of a quadrilateral?
The new idea
Read §§ 191–204: convex and concave polygons, equal polygons, mutually equiangular and mutually equilateral polygons, and the names pentagon to dodecagon.
The sum of the interior angles of a polygon is equal to two right angles, taken as many times less two as the figure has sides.
The exterior angles of a polygon, made by producing each of its sides in succession, are together equal to four right angles.
Draw the figure
Draw a hexagon ABCDEF. From A draw all the diagonals to the other vertices. Count the triangles.
Then draw a pentagon and extend each side in turn, all in the same direction round the figure.
Walk through the proof
XXXIX: from one vertex the diagonals divide an n-sided polygon into n − 2 triangles, each with angles adding to 2 right angles.
XL: at each vertex, interior + exterior = 2 right angles, so all together make 2n right angles. Take away the interior sum, (n − 2) × 2 right angles, and 4 right angles are left.
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.
Exercises
Ex. 17. How many diagonals can be drawn in a polygon of n sides ?
Ex. 18. How many sides has a polygon if the sum of its interior angles is twice the sum of its exterior angles ? ten times the sum of its exterior angles ?
Wrap up
Ask: How many degrees is each angle of a regular hexagon? (120°.)
Answers to the book's problems
Ex. 17: n(n − 3)/2. Ex. 18: 6 sides; 22 sides.
Worked out for this site from the scan. If a number in your copy differs, trust the book.