Each angle of a regular octagon has a measure of \(\boxed{{\mathbf{135^\circ }}}\) .
Further explanation:
A regular octagon is shown in Figure 1.
Now, from Figure 1 it is observed that a regular octagon has 8 sides and 8 angles.
The formula to obtain the sum of the interior angles with side \(n\) is given below.
\({\text{Sum of interior angles}}=\left({n-2}\right)180^\circ\) …… (1)
Here, regular octagon has 8 sides that is \(n=8\) .
Substitute \(8\) for \(n\) in equation (1) to obtain the sum of interior angle of regular octagon.
\(\begin{aligned}{\text{Sum of interior angles}}&=\left({8-2}\right)180^\circ\\&=\left(6\right)180^\circ\\&=1080^\circ\\\end{aligned}\)
Therefore, the sum of interior angles of a regular octagon is \(1080^\circ\) .
The measure of each interior angle with side \(n\) and sum of interior angle \(s\) is given below.
\({\text{Interior angle}}=\frac{{\text{s}}}{{\text{n}}}\)
…… (2)
Now, substitute \(8\) for \(n\) and \(1080\) for \(s\) in equation (2) to obtain the measure of each interior angle of a regular octagon.
\(\begin{aligned}{\text{interior angle}}&=\frac{{1080^\circ}}{8}\\&=135^\circ\\\end{aligned}\)
Therefore, each angle of a regular octagon is \(135^\circ\) .
Thus, each angle of a regular octagon has a measure of \(\boxed{{\mathbf{135^\circ }}}\) .
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Answer Details:
Grade: Junior High School
Subject: Mathematics
Chapter: Lines and angles
Keywords: angle, regular octagon, \(135\) degree, measure, angle of a regular octagon, \(1080\) degree, sum of interior angles.
