Question

The measure of the external angle of a regular octagon is
(a) $\dfrac{\pi }{4}$
(b) $\dfrac{\pi }{6}$
(c) $\dfrac{\pi }{8}$
(d) $\dfrac{\pi }{12}$

Answer 1

Hint: First, we should know that the summation of exterior angles of any polygon is equal to $360{}^\circ $ . So, by dividing this total $360{}^\circ $ by 8 i.e. octagon has total 8 sides, we get each exterior angle of an octagon. Then as options are in radian form, we will multiply the answer by $\dfrac{\pi }{180}$.

Complete step by step answer:
Here, we are asked to find the exterior angle of a regular polygon. So, we will simply be understood by figure of octagon which is given as below:

As we can count here, there are 8 sides of a regular octagon. We should know that the sum of the exterior angles is always equal to $360{}^\circ $ .
This is like summation of all exterior angles of 8 sides is $360{}^\circ $ then what is the angle of 1side. So, this can be solved by unitary method given as:
$\begin{align}
  & 8sides=360{}^\circ \\
 & 1side=? \\
\end{align}$
On solving this, we get as
$=\dfrac{360{}^\circ }{8}=45{}^\circ $
Thus, 1 exterior angle of a regular hexagon is $45{}^\circ $ .
Now, converting this into radian form multiplying it with $\dfrac{\pi }{180}$ . So, we get
$=45{}^\circ \times \dfrac{\pi }{180{}^\circ }=\dfrac{\pi }{4}$
Thus, angle is $\dfrac{\pi }{4}$ in radian form.
Hence, option (a) is correct.

Note: Another approach is to first find the value of each interior angle of the octagon and then to subtract it from supplementary angle i.e. $180{}^\circ $ .
So, the formula to find the interior angle of any polygon is given by $\dfrac{\left( n-2 \right)\times 180}{n}$ where n is the sides of a polygon. Here, we will take n as 8 and on solving, we get as
$=\dfrac{\left( 8-2 \right)\times 180}{8}=\dfrac{1080}{8}=135{}^\circ $
Now, we should know that each exterior angle is the supplementary angle to the interior angle at the vertex of the polygon, so each exterior angle will be $=180{}^\circ -135{}^\circ =45{}^\circ $ . On converting it to radian, we get the same answer.