Question

Determine the number of sides of a polygon whose exterior and interior angles are in the ratio 1:5.

Answer 1

Hint: To solve this first we need know that sum of the interior angle in any polygon is calculated by $ \left( {n - 2} \right) \times 180^\circ $ , where n is the number of sides of any polygon and the sum of exterior angle is equal to $ 360^\circ $ .

Complete step-by-step answer:
Given, there is a polygon whose exterior and interior angles are in the ratio 1:5.
Let, there be n sides of that polygon.
As, the sum of interior angles of any polygon is calculated by $ \left( {n - 2} \right) \times 180^\circ $ and sum of the exterior angle is equal to $ 360^\circ $ .
So,
\[
  \dfrac{{360^\circ }}{{\left( {n - 2} \right) \times 180^\circ }} = \dfrac{1}{5} \\
  360^\circ \times 5 = \left( {n - 2} \right) \times 180^\circ \\
  \dfrac{{360^\circ }}{{180^\circ }} \times 5 = n - 2 \\
\Rightarrow 2 \times 5 = n - 2 \\
\Rightarrow n - 2 = 10 \\
\Rightarrow n = 10 + 2 \\
\Rightarrow n = 12 \;
 \]
Therefore, there are 12 sides in the given polygon.
So, the correct answer is “12”.

Note: This question can be solved in a different method.
As it is given the ratio of external and internal angles 1:5.
Let, exterior angle be x, interior angle be 5x.
As, we know that the sum of interior angle and the sum of exterior angle is always equal to $ 180^\circ $ .
So,
 $
\Rightarrow x + 5x = 180^\circ \\
\Rightarrow 6x = 180^\circ \\
\Rightarrow x = 30^\circ \;
  $
So, the exterior angle is $ 30^\circ $ and the interior angle is five times exterior angle $ 5 \times 30^\circ = 150^\circ $ .
It is known that no of sides of a polygon is calculated by the formula.
 $
  {\text{No}}{\text{.}}\,{\text{of}}\,{\text{sides}}\,{\text{of}}\,{\text{a}}\,{\text{polygon}} = \dfrac{{360^\circ }}{{{\text{Interior}}\,{\text{angle}}}} \\
  {\text{No}}{\text{.}}\,{\text{of}}\,{\text{sides}}\,{\text{of}}\,{\text{a}}\,{\text{polygon}} = \dfrac{{360^\circ }}{{30^\circ }} \\
   = 12 \\
  $
So, the polygon is 12 sides.