Wentworth's Plane and Solid Geometry · Book V, Props. XXIII–XXVI · Book 6: Book V: Regular Polygons and Circles
74. The regular polygon is the greatest
Goal: Your student can prove that among polygons with given sides the inscribed one is greatest, and that among isoperimetric polygons the regular one is greatest.
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 loop of string
What's in the book
Propositions XXIII–XXVI (§§ 487–491) and Exercises 445–451.
The lesson, step by step
Warm up
Ask: State Props. XX and XXI.
The new idea
Of all polygons with given sides, that which can be inscribed in a circle is the maximum.
Of isoperimetric polygons of the same number of sides, the maximum is equilateral.
Of isoperimetric regular polygons, that which has the greatest number of sides is the maximum.
Prop. XXVI (§ 491): of regular polygons having a given area, that which has the greatest number of sides has the least perimeter.
Draw the figure
With a loop of string 24 cm long, make a triangle, a square and a hexagon on paper and estimate their areas. The figure with more sides holds more.
Walk through the proof
XXIII: draw the diameter from A; each half is a polygon with all sides given but one, so it is at least as big as the matching half of any other polygon with those sides (Prop. XXII).
XXIV: any three consecutive vertices must form the greatest triangle on their base, so it is isosceles (Prop. XXI).
XXV: cut the triangle and turn the piece over to make an irregular four-sided figure with the same perimeter and area; the regular four-sided figure beats it.
XXVI: follows from XXV.
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. 446. Of all equivalent rectangles, the square has the minimum perimeter.
Ex. 449. To inscribe in a semicircle the maximum rectangle.
Wrap up
Read Ex. 555 and 556 (page 247) and ask what they say about the circle. (It is the "best" figure of all.)
Answers to the book's problems
Ex. 446: use Ex. 445 and the fact that the square is the regular four-sided figure. Ex. 449: it is half of the square inscribed in the whole circle: its base on the diameter is R√2 and its height R√2/2.
Worked out for this site from the scan. If a number in your copy differs, trust the book.