Question

Find the measure of each angle (in degrees) of a regular octagon.
(a) ${{90}^{\circ }}$
(b) ${{60}^{\circ }}$
(c) ${{135}^{\circ }}$
(d) ${{145}^{\circ }}$

Answer 1

Hint: In order to find the measure of each angle of a regular octagon in degrees, we need to remember the formula to calculate each of the interior angle of a regular polygon, which is given by the following formula:
Interior angle of a regular polygon $=\dfrac{\left( 2n-4 \right)\times {{90}^{\circ }}}{n}.............(i)$where n is the number of sides of a regular polygon. In equation (i), if we replace n with the number of sides of a regular polygon, we can easily obtain the measure of each of the interior angles of a regular octagon in degrees.

Complete step-by-step answer:
Here, we have to find the measure of each angle of a regular octagon in degrees. We know, there are eight sides in a regular octagon as shown in the figure. This implies that we have to replace the value of ‘n’ with 8 in the above equation (i).

On replacing the value of n, we will get the required measure of each angle of a regular octagon in degrees.
We know, number of sides in a regular octagon, $n=8$.
On replacing the value of ‘n’ in equation (i) and solving the equation, we get,
Each angle \[=\dfrac{\left( 2\times 8-4 \right)\times {{90}^{\circ }}}{8}\]
           \[=\dfrac{\left( 16-4 \right)\times {{90}^{\circ }}}{8}\]
          \[=\dfrac{12\times {{90}^{\circ }}}{8}\]
          \[=\dfrac{3\times {{90}^{\circ }}}{2}\]
          \[=3\times {{45}^{\circ }}\]
            \[={{135}^{\circ }}\]
Hence, the measure of each angle of a regular octagon is ${{135}^{\circ }}$.

So, the correct answer is “Option C”.

Note: Students normally make mistakes in replacing the value of n. Students have confusion with the number of sides present in a regular octagon. They generally assume an octagon has 6 or 7 or 8 numbers of sides. This mistake leads to wrong calculation and thereby to mistake in the entire question. So, it's very important that students learn about each of the regular polygons, the number of sides they have and also the shape. Besides, it is very important that the value of ‘n’ is replaced properly in the correct formula. Students need to remember the formula to calculate each exterior angle of a given polygon as this formula can be used in any other question, where all we need to do is replace the value of ‘n’ correctly.