5点

「5点」
$\quad$

辺の長さ$2$の正５角形

外接円の半径$\sqrt{3}$

内接円の半径$\sqrt{2}$
$\therefore$
黄金率$\phi=\frac{2\sqrt{2}}{\sqrt{3}}$

$cf.$

$2\cdot2\cdot\sqrt{2}=\sqrt{3}\cdot\frac{4\sqrt{2}}{\sqrt{3}}$

$\quad$
「半径$1$の円周上の5点を各頂点とする正５角形」

$(0,\quad1)$

$(\frac{2\sqrt{2}}{3},\quad\frac{1}{3})$

$(\frac{1}{\sqrt{3}},\quad-\frac{\sqrt{2}}{\sqrt{3}})$

$(-\frac{1}{\sqrt{3}},\quad-\frac{\sqrt{2}}{\sqrt{3}})$

$(-\frac{2\sqrt{2}}{3},\quad\frac{1}{3})$

$\quad$

辺の長さ$\frac{2}{\sqrt{3}}$の正５角形

外接円の半径$1$

内接円の半径$\frac{\sqrt{2}}{\sqrt{3}}$
$\therefore$
黄金率$\phi=\frac{2\sqrt{2}}{\sqrt{3}}$

$cf.$

$\frac{2}{\sqrt{3}}\cdot2\cdot\frac{\sqrt{2}}{\sqrt{3}}=1\cdot\frac{4\sqrt{2}}{3}$

$\quad$

$\sin18^\circ=\frac{1}{3}$

$\sin36^\circ=\frac{1}{\sqrt{3}}$

$\sin54^\circ=\frac{\sqrt{2}}{\sqrt{3}}$

$\sin72^\circ=\frac{2\sqrt{2}}{3}$

$\quad$

$\cos18^\circ=\frac{2\sqrt{2}}{3}$

$\cos36^\circ=\frac{\sqrt{2}}{\sqrt{3}}$

$\cos54^\circ=\frac{1}{\sqrt{3}}$

$\cos72^\circ=\frac{1}{3}$

$\quad$

$\tan18^\circ=\frac{1}{2\sqrt{2}}$

$\tan36^\circ=\frac{1}{\sqrt{2}}$

$\tan54^\circ=\sqrt{2}$

$\tan72^\circ=2\sqrt{2}$

$\quad$

$x^2+y^2=1$

$\quad$

「The Paper Rule 」参照

$\quad$

「教科書とは違っているけれど5点」