Maximizing area of a pentagon

Question

Suppose $a,b,c,d,e$ are pairwise distinct positive integers. Consider a pentagon with sides $a,b,c,d,e$ and with angles maximizing its area (we assume that a pentagon with such sides exists). It is easy to see that its area $S$ is a positive algebraic number.

What is the smallest possible degree of $S$? Can it be a quadratic irrational? Can it be an integer?

Answer 1

Yes, the area $S$ can be an integer.

Given any set of positive numbers $a_1, a_2, \ldots, a_n$. It is known that among the $n$-gons having this set as side lengths, the one maximizing the area is cyclic and is unique up to ordering of the sides.

Our problem reduces to finding a cyclic pentagon whose side lengths and area are all integers.

Consider the pentagon $ABCDE$ with vertices $A,B,C,D,E$ at: $$\small \left(\frac{325}{2},0\right), \left(\frac{3713}{26},\frac{1008}{13}\right), \left(-\frac{91}{2},156\right), \left(-\frac{2047}{26},\frac{1848}{13}\right), \left(-\frac{2975}{26},-\frac{1500}{13}\right) $$ This pentagon is cyclic. Its vertices lie on a circle centered at origin with radius $\frac{325}{2}$. In addition, its side lengths and area are all integers: $$(AB, BC, CD, DE, EA ) = ( 80, 204, 36, 260, 300 )\quad\text{ and }\quad S = 44160$$

To those who wonder how to find such a pentagon. The basic idea is working with integers which have multiple inequivalent representation as sum of two squares. If you identify the euclidean plane with the complex plane, the vertices of above pentagon have following factorization over Gaussian integers.

$$\begin{cases} 26A &= (1+2i)^2(1-2i)^2(2+3i)^2(2-3i)^2\\ 26B &= (1+2i)^4(1-2i)^0(2+3i)^0(2-3i)^4\\ 26C &= (1+2i)^0(1-2i)^4(2+3i)^2(2-3i)^2\\ 26D &= (1+2i)^4(1-2i)^0(2+3i)^4(2-3i)^0\\ 26E &= (1+2i)^2(1-2i)^2(2+3i)^4(2-3i)^0 \end{cases} $$

Update

It turns out this sort of cyclic pentagons with rational side lengths and area has a name! It is known as Robbins pentagon. It is named after David P. Robbins who had given a formula for the area of a cyclic pentagon as a function of its sides${}^{\color{blue}{[1]}}$.

Consider a cyclic pentagon with sides $a_1, \ldots, a_5$ and area $S$. If $\sigma_1, \ldots, \sigma_5$ are the symmetric polynomials in the squares of the sides, then $u = 16S^2$ satisfies a degree $7$ condition $$u t_4^3 + t_3^2 t_4^2 - 16 t_3^3 t_5 - 18u t_3 t_4 t_5 - 27 u^2t_5^2 = 0 \quad\text{ where }\quad \begin{cases} t_2 &= u − 4\sigma_2 + \sigma_1^2\\ t_3 &= 8\sigma_3 + \sigma_1 t_2\\ t_4 &= -64\sigma_4 + t_2^2\\ t_5 &= 128\sigma_5 \end{cases} $$

A consequence of this is the area $S$ of any cyclic pentagon with integer sides is algebraic with degree at most $14$.

According to a paper ${}^{\color{blue}{[2]}}$ by MacDougall and Buchholz, there are other cyclic pentagons with integer sides and area. Following is a short list for pentagons with $S \le 3000$.

peri-
meter   sides        radius  area diagonals
 68 [7,7,15,15,24]     25/2   276 [336/25,20,24,117/5,25]
 72 [7,15,15,15,20]    25/2   342 [20,24,24,25,117/5]
178 [9,20,20,51,78]   325/8  1332 [143/5,504/13,65,1161/25,75]
172 [16,16,25,52,63]   65/2  1638 [2016/65,39,63,253/5,65]
176 [16,25,33,39,63]   65/2  1848 [39,52,60,60,65]
178 [16,25,25,52,60]   65/2  1884 [39,600/13,63,56,836/13]
182 [16,25,33,52,56]   65/2  2058 [39,52,323/5,60,312/5]
182 [25,25,33,39,60]   65/2  2094 [600/13,52,60,63,65]
184 [16,25,39,52,52]   65/2  2148 [39,56,65,312/5,60]
186 [25,33,33,39,56]   65/2  2268 [52,3696/65,60,323/5,837/13]
188 [25,33,39,39,52]   65/2  2358 [52,60,312/5,65,63]
238 [12,12,55,55,104] 325/6  2424 [7752/325,65,1232/13,371/5,100]
218 [13,13,40,68,84]   85/2  2436 [2184/85,51,84,304/5,85]
220 [9,20,51,65,75]   325/8  2760 [143/5,65,406/5,70,78]
220 [20,20,51,51,78]  325/8  2844 [504/13,65,25806/325,75,406/5]
224 [13,36,40,51,84]   85/2  2856 [805/17,68,77,75,85]
224 [9,20,65,65,65]   325/8  2952 [143/5,75,78,78,70]

References

• $\color{blue}{[1]}$ Robbins, David P. (1994), Areas of polygons inscribed in a circle, Discrete and Computational Geometry 12 (2) 223–236

• $\color{blue}{[2]}$ MacDougall, James A. and Buchholz, Ralph H. (2008) Cyclic Polygons with Rational Sides and Area. Journal of Number Theory, 128 (1). pp. 17-48. ( an online copy can be found here )