[tex]\text{Dlugosc okregu: } l = 2\pi r\\\text{Dlugosc wycinka kola: } l_w=\frac{\alpha}{360^\circ}*2\pi r\\\\\text{Pole kola: } P=\pi r^2\\\text{Pole wycinka kola: } P_w=\frac{\alpha}{360^\circ}*\pi r^2[/tex]
[tex]x=\frac{\alpha}{360^\circ}\\\\P_w=12\pi\\l_w=4\pi\\\\\left \{ {{x \pi r^2=12\pi} \atop {2x\pi r=4\pi}} \right. \\\left \{ {{x r^2=12} \atop {xr=2 \to r=\frac{2}x}} \right. \\\left \{ {{x*(\frac{2}x)^2=12} \atop {r=\frac{2}x}} \right. \\\left \{ {{x*\frac{4}{x^2}=12} \atop {r=\frac2x}} \right. \\\left \{ {{\frac{4}x=12 \to \frac4{12}=x} \atop {r=\frac{2}{\frac4{12}}}} \right. \\\left \{ {{x=\frac13} \atop {r=6}} \right.[/tex]
[tex]\frac13=\frac{\alpha}{360^\circ}\\3\alpha=360^\circ /:3\\\underline{\alpha=120^\circ}[/tex]
