# 五边形本文重定向自 五边形

5

${\displaystyle \approx 1.720477400589a^{2}}$

## 正五边形

${\displaystyle ={\frac {\sqrt {5+2{\sqrt {5}}}}{2}}\cdot }$边长${\displaystyle \approx 1.539\cdot }$边长
${\displaystyle ={\frac {1+{\sqrt {5}}}{2}}\cdot }$边长${\displaystyle \approx 1.618\cdot }$边长

${\displaystyle A={\frac {t^{2}{\sqrt {25+10{\sqrt {5}}}}}{4}}={\frac {5t^{2}\tan(54^{\circ })}{4}}\approx 1.720t^{2}.}$

### 面积公式推导

${\displaystyle A={\frac {1}{2}}Pr}$

${\displaystyle A={\frac {1}{2}}\times 5t\times {\frac {t\tan(54^{\circ })}{2}}={\frac {5t^{2}\tan(54^{\circ })}{4}}}$

### 内切圆半径

${\displaystyle r={\frac {t}{2\tan(\pi /5)}}={\frac {t}{2{\sqrt {5-{\sqrt {20}}}}}}\approx 0.6882\cdot t}$

### 构造

${\displaystyle \tan(\phi /2)={\frac {1-\cos(\phi )}{\sin(\phi )}}\ ,}$

${\displaystyle h={\frac {{\sqrt {5}}-1}{4}}\ .}$

${\displaystyle a^{2}=1-h^{2}\ ;\ a={\frac {1}{2}}{\sqrt {\frac {5+{\sqrt {5}}}{2}}}\ .}$

${\displaystyle s^{2}=(1-h)^{2}+a^{2}=(1-h)^{2}+1-h^{2}=1-2h+h^{2}+1-h^{2}=2-2h=2-2\left({\frac {{\sqrt {5}}-1}{4}}\right)\ }$
${\displaystyle ={\frac {5-{\sqrt {5}}}{2}}\ .}$

${\displaystyle s={\sqrt {\frac {5-{\sqrt {5}}}{2}}}\ ,}$

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