\[n\]
sided regular polygon with the interior and exterior angles in the ratio \[11:1\]
.
\[11:1\]
.\[\frac{180}{11+1}=15\]
degrees.The interior angle is
\[11 \times 15 = 165\]
degrees and the exterior angle is \[1 \times 15-15\]
degrees.As the perimeter of the polygon is traced out, an angle of 15 degrees is turned through at each vertex.
\[n\]
of these turns are made, adding up to a complete turn of 360 degrees. Hence \[n=\frac{360}{15}=24\]
.The polygon has 24 sides.