Wentworth's Plane and Solid Geometry · Book V, §§ 482–486 (Props. XX–XXII) · Book 6: Book V: Regular Polygons and Circles

73. Maxima and minima: triangles and polygons

Goal: Your student understands maximum and minimum and can prove the first three theorems on figures of greatest area.

⏱ About 40 minutes · 6 steps

Open it at the Internet Archive ↗

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)
  • String and two pins

What's in the book

Definitions of maximum, minimum and isoperimetric polygons (§§ 482–483), and Propositions XX–XXII (§§ 484–486).

The lesson, step by step

  1. Warm up

    Ask: What is the largest chord in a circle? The shortest line from a point to a line?

  2. The new idea

    Of all triangles having two given sides, that in which these sides include a right angle is the maximum.

    Of all isoperimetric triangles having the same base the isosceles triangle is the maximum.

    Of all polygons with sides all given but one, the maximum can be inscribed in a semicircle which has the undetermined side for its diameter.

  3. Draw the figure

    For Prop. XX, use two pencils hinged at one end as the two given sides: open them to different angles and imagine the triangle. Which opening seems biggest?

    For Prop. XXI, tie a loop of string around two pins (the base) and stretch it with a pencil.

  4. Walk through the proof

    XX: with one side as base, the area depends on the altitude, which is greatest when the other side stands perpendicular.

    XXI: compare the altitudes of the two triangles using the auxiliary lines in the book; the isosceles triangle's altitude is greater.

    XXII: if the polygon were largest but some angle at a vertex were not right, we could enlarge it by Prop. XX. So each vertex sees the free side under a right angle, and lies on the semicircle.

  5. 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.

  6. Wrap up

    Ask: A farmer has a fixed length of fence and a straight wall. What shape encloses the most? (Prop. XXII suggests a half-circle against the wall.)

@teddyrookbook GitHub