Determine the number of sides of a polygon whose exterior

Question:

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

Solution:

Let $n$ be the number of sides of a polygon.

Let $x$ and $5 x$ be the exterior and interior angles.

Since the sum of an interior and the corresponding exterior angle is $180^{\circ}$, we have:

$x+5 x=180^{\circ}$

$\Rightarrow 6 x=180^{\circ}$

$\Rightarrow x=30^{\circ}$

The polygon has $n$ sides.

So, sum of all the exterior angles $=(30 \mathrm{n})^{\circ}$

We know that the sum of all the exterior angles of a polygon is $360^{\circ}$.

i.e., $30 n=360$

$\therefore n=12$

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