Rysunek pomocniczy prostokąta:

Korzystając z tw. cosinusów otrzymujemy:

$a^2=6^2+6^2-2\cdot 6\cdot 6\cdot \cos {30}^{\circ }$  

$a^2=36+36-72\cdot \frac{\sqrt{3}}{2}$  

$a^2=72-36\sqrt{3}$  

$a^2=36\left(2-\sqrt{3}\right)$  

$a=6\sqrt{2-\sqrt{3}}$  

Obliczmy długość boku b, korzystając z tw. Pitagorasa.

$a^2+b^2={12}^2$  

$72-36\sqrt{3}+b^2=144\mid -72+36\sqrt{3}$  

$b^2=72+36\sqrt{3}$  

$b^2=36\left(2+\sqrt{3}\right)$  

$b=6\sqrt{2+\sqrt{3}}$  

 

Przypadek I.

$h=6\sqrt{2-\sqrt{3}}$  

$2\pi r=6\sqrt{2+\sqrt{3}}\mid :2$  

$\pi r=3\sqrt{2+\sqrt{3}}{\mid }^2$  

${\pi }^2{r}^2=9\left(2+\sqrt{3}\right)$  

$r^2=\frac{9\left(2+\sqrt{3}\right)}{{\pi }^2}$  

 

Obliczmy objętość tego walca.

$V=\pi r^{2}h$  

$V=\pi \cdot \frac{9\left(2+\sqrt{3}\right)}{{\pi }^2}\cdot 6\sqrt{2-\sqrt{3}}$  

$V=\frac{54}{\pi }\left(2+\sqrt{3}\right)\sqrt{2-\sqrt{3}}$  

$V=\frac{54}{\pi }\cdot \sqrt{{\left(2+\sqrt{3}\right)}^2\cdot \left(2-\sqrt{3}\right)}$  

$V=\frac{54}{\pi }\cdot \sqrt{\left(4+4\sqrt{3}+3\right)\cdot \left(2-\sqrt{3}\right)}$  

$V=\frac{54}{\pi }\cdot \sqrt{\left(7+4\sqrt{3}\right)\left(2-\sqrt{3}\right)}$  

$V=\frac{54}{\pi }\cdot \sqrt{14-7\sqrt{3}+8\sqrt{3}-12}$  

$V=\frac{54}{\pi }\cdot \sqrt{2+\sqrt{3}}\left[{\text{dm}}^3\right]$  

 

Przypadek II.

$h=6\sqrt{2+\sqrt{3}}$  

$2\pi r=6\sqrt{2-\sqrt{3}}\mid :2$  

$\pi r=3\sqrt{2-\sqrt{3}}{\mid }^2$  

${\pi }^2{r}^2=9\left(2-\sqrt{3}\right)$  

$r^2=\frac{9\left(2-\sqrt{3}\right)}{{\pi }^2}$  

 

Obliczmy objętość tego walca.

$V=\pi r^{2}h$  

$V=\pi \cdot \frac{9\left(2-\sqrt{3}\right)}{{\pi }^2}\cdot 6\sqrt{2+\sqrt{3}}$  

$V=\frac{54}{\pi }\cdot \left(2-\sqrt{3}\right)\cdot \sqrt{2+\sqrt{3}}$  

$V=\frac{54}{\pi }\left(2-\sqrt{3}\right)\sqrt{2+\sqrt{3}}$  

$V=\frac{54}{\pi }\cdot \sqrt{{\left(2-\sqrt{3}\right)}^2\cdot \left(2+\sqrt{3}\right)}$  

$V=\frac{54}{\pi }\cdot \sqrt{\left(4-4\sqrt{3}+3\right)\cdot \left(2+\sqrt{3}\right)}$  

$V=\frac{54}{\pi }\cdot \sqrt{\left(7-4\sqrt{3}\right)\left(2+\sqrt{3}\right)}$  

$V=\frac{54}{\pi }\cdot \sqrt{14+7\sqrt{3}-8\sqrt{3}-12}$  

$V=\frac{54}{\pi }\cdot \sqrt{2-\sqrt{3}}\left[{\text{dm}}^3\right]$