A small polygon is a polygon of unit diameter. The question of finding the largest area of small n-gons has been answered for some values of n. Regular n-gons are optimal when n is odd and kites with unit length diagonals are optimal when \(n=4\). For \(n=6\), the largest area is a root of a degree 10 polynomial with integer coefficients and height 221360 (the height of a polynomial is the largest coefficient in absolute value). The present paper analyses the and octogonal cases, and under an axial symmetry conjecture, we propose a methodology that leads to a polynomial of degree 344 with integer coefficients that factorizes into a polynomial of degree 42 with height 23588130061203336356460301369344. A root of this last polynomial corresponds to the area of the largest small axially symmetrical octagon.

小多边形是单位直径的多边形。对于某些n值，已经回答了寻找最大的小n边形区域的问题。当n为奇数时，常规n边形为最佳，当\（n = 4 \）时，具有单位长度对角线的风筝为最佳。对于\（n = 6 \），最大面积是具有整数系数和高度221360的10度多项式的根（多项式的高度是绝对值中的最大系数）。本文分析了正交和八角形的情况，并在轴向对称猜想下，提出了一种方法，该方法可导致整数系数为344的多项式分解为高度为23588130061203336356460460301369344的42多项式的分解。到最大的轴向对称小八边形的面积。