Question

The sum of the interior angles of a polygon is three times the sum of its exterior angles. Determine the number of sides of the polygon.

Answer 1

Hint: In this question use the concept that sum of exterior angle of a polygon is ${360^ \circ }$ and the sum of interior angle of a polygon is $\left( {n - 2} \right) \times {180^ \circ }$ where n is the number of sides of a polygon.

Complete step-by-step answer:

As we know the sum of interior angles of a regular polygon $ = \left( {n - 2} \right) \times {180^ \circ }$, where n is the number of sides. 

Now as we know that the sum of exterior angles of a regular polygon $ = {360^ \circ }$

Now it is given that the sum of interior angles of a polygon is three times the sum of exterior angles.

Therefore sum of interior angles of a regular polygon = $3 \times $ sum of exterior angles.

$ \Rightarrow \left( {n - 2} \right) \times {180^ \circ } = 3 \times {360^ \circ }$

Now simplify the above equation we have,

$ \Rightarrow \left( {n - 2} \right) = 3 \times \dfrac{{{{360}^ \circ }}}{{{{180}^ \circ }}} = 6$

$ \Rightarrow n = 6 + 2 = 8$

Therefore the sides of the regular polygon = 8.

Note: The basic understanding of Interior angle and exterior angle is the key part of this problem. Interior angle is the angle of a polygon inside of it at one of its vertices whereas exterior angle is an acute angle outside the polygon formed by one of its sides and the extension of the adjacent side.