Chapter 13.4, Problem 7MRE

To determine

To find the each exterior angle of a regular octagon.

Answer to Problem 7MRE

Each exterior angle of a regular octagon is $\frac{5\pi }{4}$ .

Explanation of Solution

Given information: Each exterior angle of a regular octagon.

Calculation: The required angle of a regular octagon is obtained as,

Hexagon figure is,

  

O is the center.

  $\angle AOB=\frac{2\pi }{8}=\frac{\pi }{4}$

In triangle AOB,

  $\begin{array}{l}\angle OAB=\angle ABO\\ \therefore 2\angle OAB+\angle AOB=\pi \\ 2\angle OAB+\frac{\pi }{4}=\pi \\ \angle OAB=\frac{3\pi }{8}\end{array}$

Interior angle of octagon,

  $\begin{array}{l}\angle ABC=\angle ABO+\angle OBC\\ =2\angle ABO\\ =2\times \frac{3\pi }{8}\\ =\frac{3\pi }{4}\end{array}$

Therefore, interior angle of octagon is $\frac{3\pi }{4}$ .

Exterior angle of octagon is,

  $2\pi -\frac{3\pi }{4}=\frac{5\pi }{4}$

Hence, each exterior angle of a regular octagon is $\frac{5\pi }{4}$ .