# 五边形

## 维基百科，自由的百科全书

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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## 参考文献

1. Exhaustive search of convex pentagons which tile the plane
2. ^ Herbert W Richmond. Pentagon. 1893.
3. ^ Peter R. Cromwell. Polyhedra. : 63. ISBN 0-521-66405-5.
4. ^ This result agrees with Herbert Edwin Hawkes; William Arthur Luby; Frank Charles Touton. Exercise 175. Plane geometry. Ginn & Co. 1920: 302.
5. ^ H.S.M. Coxeter Regular Polytopes, 3rd edition, 1973
6. ^ Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
7. ^ (Paper 10) H.S.M. Coxeter, Star Polytopes and the Schlafli Function f(α,β,γ) [Elemente der Mathematik 44 (2) (1989) 25–36]
8. ^ Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Table I(ii): 16 regular polytopes {p, q, r} in four dimensions, pp. 292–293)