# 五邊形

5

4
a2cotπ
5

1.720477400589a2

## 正五邊形

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

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

### 面積公式推導

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

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

### 內切圓半徑

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

### 構造

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

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

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

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

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

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## 參考文獻

1. Peralta, Eyder. With Discovery, 3 Scientists Chip Away At An Unsolvable Math Problem. NPR. 2015-08-14 [2015-08-15]. （原始內容存檔於2016-08-01）.
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 , 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)