Question

The exterior angle of a regular polygon is one-third of its interior angle. Determine how many sides does the polygon has.
(a) 10
(b) 8
(c) 9
(d) 13

Answer 1

Hint: In this question, in order to determine the number of sides of the regular polygon whose exterior angle of the polygon being one-third of the interior angle, we will assume that the interior angle of the regular polygon is given by \[x\]. Then the exterior angle of the polygon is given by \[\dfrac{1}{3}x\]. Now we know the exterior angle property for a regular polygon which says that the sum of the adjacent interior and exterior angle is \[{{180}^{\circ }}\]. Thus we will get an equation \[\dfrac{1}{3}x+x={{180}^{\circ }}\]. Solving this will give the value of the interior angle and then the exterior angle can be calculated accordingly. Thus the number of sides of the regular polygon can be calculated using \[\dfrac{360}{\text{exterior}\,\text{angle}}\].

Complete step-by-step solution:
A regular polygon is a polygon whose sides are of equal length.
Let us suppose that the interior angle of the regular polygon is given by \[x\].
Then the exterior angle of the polygon being one-third of the interior angle of the same regular polygon is given by \[\dfrac{1}{3}x\].
Now we know the exterior angle property for a regular polygon which says that the sum of the adjacent interior and exterior angle is \[{{180}^{\circ }}\].
Using this we get that
\[\dfrac{1}{3}x+x={{180}^{\circ }}\]
On solving the above equation to find the value of \[x\], we get
\[x\left( \dfrac{1}{3}+1 \right)={{180}^{\circ }}\]
\[\Rightarrow x\left( \dfrac{4}{3} \right)={{180}^{\circ }}\]
\[\Rightarrow x=\dfrac{3}{4}\times {{180}^{\circ }}\]
\[\Rightarrow x=3\times {{45}^{\circ }}\]
\[\Rightarrow x={{135}^{\circ }}\]
Thus, the interior angle of the regular polygon is given by \[{{135}^{\circ }}\].
Thus, the exterior angle of the regular polygon is given by \[\dfrac{{{135}^{\circ }}}{3}={{45}^{\circ }}\].
Now the number of sides of the regular polygon is given by \[\dfrac{360}{\text{exterior}\,\text{angle}}\].
Substituting the value of the exterior angle of the given polygon in the above formula we get,
\[\dfrac{360}{45}=8\]
Therefore the given regular polynomial has 8 sides. Hence it is a regular octagon.
Thus option (b) is correct.

Note: In this problem, do not calculate the number of sides by dividing 360 by the interior angle of the regular calculation which is a very trivial mistake. Follow exact properties like the exterior angle property and the formula to count the number of sides of the regular polynomial with the given piece of information.