The Notes: In a regular polygon the interior angles are 36 degrees greater than the exterior angles. How many sides does this polygon have?

Monday, March 2, 2009

In a regular polygon the interior angles are 36 degrees greater than the exterior angles. How many sides does this polygon have?

Hello!

The interior and exterior angles are supplementary ones, thus the sum of their measures is $\displaystyle{180}$ degrees. So if we denote an exterior angle measure as $\displaystyle{x},$ then the interior measure is $\displaystyle{180}-{x}.$ It is also given that "the interior angles are 36 degrees greater than the exterior angles",

$\displaystyle{180}-{x}={36}+{x},$

therefore $\displaystyle{x}=\frac{{{180}-{36}}}{{2}}={72}$ (degrees).

For a regular polygon with $\displaystyle{n}$ angles (and $\displaystyle{n}$ sides, of course), an exterior angle has the measure of $\displaystyle\frac{{360}}{{n}}.$ So we have