十边形

10

t{5}

${\displaystyle \approx 7.694208842938a^{2}}$

正十边形

面积

${\displaystyle A={\frac {5}{2}}a^{2}\cot \left({\frac {\pi }{10}}\right)={\frac {5}{2}}a^{2}{\sqrt {5+2{\sqrt {5}}}}\simeq 7.694208843\,a^{2}}$

${\displaystyle A=10\tan \left({\frac {\pi }{10}}\right)r^{2}=5r^{2}{\sqrt {5-2{\sqrt {5}}}}\simeq 3.249196963\,r^{2}}$

${\displaystyle A={\frac {5}{2}}\sin \left({\frac {\pi }{5}}\right)R^{2}={\frac {5}{2}}R^{2}{\sqrt {\frac {5-{\sqrt {5}}}{2}}}\simeq 2.938926262\,r^{2}}$

正十边形中的黄金比例

${\displaystyle {\frac {\overline {AM}}{\overline {MH}}}={\frac {\overline {AH}}{\overline {AM}}}={\frac {1+{\sqrt {5}}}{2}}=\Phi \approx 1.618}$
• 在已知边长构造正十边形[6]的过程中，圆弧D半径DA与线段E10F的比为黄金比例
${\displaystyle {\frac {\overline {E_{1}E_{10}}}{\overline {E_{1}F}}}={\frac {\overline {E_{10}F}}{\overline {E_{1}E_{10}}}}={\frac {R}{a}}={\frac {1+{\sqrt {5}}}{2}}=\Phi \approx 1.618{\text{.}}}$

扭歪十边形

{5}#{ } {5/2}#{ } {5/3}#{ }

 正十二面体 正二十面体 截半二十面体 菱形三十面体

参考文献

1. Sidebotham, Thomas H., The A to Z of Mathematics: A Basic Guide, John Wiley & Sons: 146, 2003, ISBN 9780471461630.
2. ^ Wenninger, Magnus J., Polyhedron Models, Cambridge University Press: 9, 1974, ISBN 9780521098595.
3. ^ The elements of plane and spherical trigonometry, Society for Promoting Christian Knowledge: 59, 1850
4. ^ Dixon, R. Mathographics. New York: Dover, p. 18, 1991. ISBN 978-0486266398
5. Green, Henry, Euclid's Plane Geometry, Books III–VI, Practically Applied, or Gradations in Euclid, Part II, London: Simpkin, Marshall,& CO.: 116, 1861. Retrieved 10 February 2016.
6. Köller, Jürgen, Regelmäßiges Zehneck, → 3. Section "Formeln, Ist die Seite a gegeben ...", 2005. Retrieved 10 February 2016.