## 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.