Appendix 1
THEOREMS FOR PRINCESS DIDO
Being a princess isn’t all it’s cracked up to be.
—Princess Diana
We shall state two theorems that undergird Princess Dido’s “solution” discussed in Chapter 1: (i) the isoperimetric property of the circle (without proof), and (ii) the “same area” theorem (with proof). The word isoperimetric means “constant perimeter,” and here concerns the largest area that can be enclosed by a closed curve of constant length. We shall be a little more formal in our language for the statement of these results.
Theorem 1 (the isoperimetric theorem): Among all planar figures of equal perimeter, the circle (and only the circle) has maximum area.
There is also an equivalent “dual” statement:
Theorem 2 (the same area theorem): Among all planar figures of equal area, the circle (and only the circle) has minimum perimeter.
This theorem is a consequence of Theorem 1, as we can show. Consider first a closed curve C of perimeter L enclosing a domain of area A. Let r be the radius of a circle of perimeter L: then L = 2πr. This encloses a disk of area πr2. By the isoperimetric property of the circle, the area enclosed by C cannot exceed πr2, and it equals πr2 if and only if C is a circle. On the other hand, πr2 = L2/4π, so we obtain the isoperimetric inequality A ≤ L2/4π, with the equality holding only for a circle. Now we are ready for the proof of Theorem 2 assuming the isoperimetric theorem. Consider next an arbitrary planar figure with perimeter L and area A. Let D be a circular disk with the same area A and perimeter l: then A = l2/4π. If l > L, the isoperimetric inequality would be violated, so L ≥ l.
To complete the argument for Princess Dido and the semicircular area discussed in Chapter 1, we can examine a generalization of this problem. Suppose that a curve of length L is now attached to a line segment of fixed length l, forming a closed curve. What should be the shape of L in order for the area enclosed to be a maximum? Consider two situations, the first in which L is a circular arc (enclosing an area Ac), and the second in which it is not (and enclosing an area A). Imagine also that in both cases a circular arc below the line segment l is added to give an additional segment of area a. By the isoperimetric theorem, A + a ≤ Ac + a, so that A ≤ Ac; clearly Princess Dido knew what she was doing!