四十八角形
{\displaystyle {\begin{aligned}S&=12a^{2}\cot {\frac {\pi }{48}}\\&=12a^{2}\left(2+{\sqrt {3}}+{\sqrt {8+4{\sqrt {3}}}}+{\sqrt {16+8{\sqrt {3}}+2{\sqrt {104+60{\sqrt {3}}}}}}\right)\\&=12a^{2}\left(2+{\sqrt {3}}+({\sqrt {6}}+{\sqrt {2}})+{\sqrt {16+8{\sqrt {3}}+10{\sqrt {2}}+6{\sqrt {6}}}}\right)\\&=12a^{2}\left(2+{\sqrt {3}}+({\sqrt {6}}+{\sqrt {2}})+2{\sqrt {4+2{\sqrt {3}}+{\sqrt {26+15{\sqrt {3}}}}}}\right).\end{aligned}}}
cos  ( 2 π / 48 ) {\displaystyle \cos(2\pi /48)} を有理数と平方根で表すことが可能である。
cos  2 π 48 = cos  π 24 = cos  ( 7.5 ∘ ) = 1 2 2 + 2 + 3 = 1 4 8 + 2 6 + 2 2 {\displaystyle \cos {\frac {2\pi }{48}}=\cos {\frac {\pi }{24}}=\cos \left(7.5^{\circ }\right)={\frac {1}{2}}{\sqrt {2+{\sqrt {2+{\sqrt {3}}}}}}={\frac {1}{4}}{\sqrt {8+2{\sqrt {6}}+2{\sqrt {2}}}}}