边长为 $a$ 的正多边形，其面积如公式 $(1)$ 所示。

$$ \begin{align} S_{n} & = \frac {1}{2} \cdot a \cdot \frac {a} {2 \tan {\frac{\pi}{n}}} \cdot n \\ & = \frac {n}{4 \tan {\frac {\pi}{n}}} \cdot a^2 \end{align} \tag{1} $$

当边长 $a =$ 1.39 时，则：

• $S_{正三角形} = \frac{3}{4 \tan \frac {\pi} {3}} a^2 = \frac{\sqrt{3}}{4} a^2 = 0.8366 $

• $S_{正方形} = \frac{4}{4 \tan \frac {\pi} {4}} a^2 = a^2 = 1.9321$

• $S_{正五边形} = \frac{5}{4 \tan \frac {\pi} {5}} a^2 = \frac{5}{4\sqrt{5-2\sqrt{5}}} a^2 = 3.3241$

• $S_{正六边形} = \frac{6}{4 \tan \frac {\pi} {6}} a^2 = \frac{3\sqrt{3}}{2} a^2 = 5.0197$

• $S_{正七边形} = \frac{7}{4 \tan \frac {\pi} {7}} a^2 = 7.0211$