## Interior and External Angles of a Regular Polygon In a Given Ratio

Suppose we have an\[n\]

sided regular polygon with the interior and exterior angles in the ratio \[11:1\]

.The interior and exterior angles add up to 180 degrees, so we have to divide 180 in the ratio

\[11:1\]

.\[\frac{180}{11+1}=15\]

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.