Question

What are the measures of an interior angle of an 11- gon?

Answer 1

Hint: First of all understand the meaning of an 11- gon. Now, consider this a regular polygon with each side equal in length. Using this fact consider all the angles of this polygon equal to each other. Apply the formula for the sum of all the interior angles of an n – sided polygon given as $\left( n-2 \right)\times {{180}^{\circ }}$ and substitute n = 11. Divide this sum of angles obtained by 11 to get the measure of each angle.

Complete step-by-step solution:
Here we have been asked to find the measure of an interior angle of an 11- gon. First we need to understand the meaning of an 11- gon.
In mathematics, the term 11- gon is used for a polygon having 11 sides. Now, we have to consider this polygon as a regular polygon whose all the 11 sides are equal in length. If all the sides are equal in length then all the angles become equal. Therefore, an 11 – gon will have 11 equal angles.
Now, the sum of all the interior angles of an n – sided polygon is given as $\left( n-2 \right)\times {{180}^{\circ }}$, so substituting n = 11 we get,
$\Rightarrow $ Sum of interior angles = $\left( 11-2 \right)\times {{180}^{\circ }}$
$\Rightarrow $ Sum of interior angles = ${{1620}^{\circ }}$
Since there are 11 equal angles so assuming the measure of each angle as x we have,
$\begin{align}
  & \Rightarrow 11x={{1620}^{\circ }} \\
 & \Rightarrow x=\dfrac{{{1620}^{\circ }}}{11} \\
 & \therefore x={{147.27}^{\circ }} \\
\end{align}$
Hence, the measure of each interior angle of the 11- gon is 147.27 degrees.

Note: Always remember the formula $\left( n-2 \right)\times {{180}^{\circ }}$ for the sum of interior angles of n – sided polygon. Note that in the above solution we weren’t provided that the polygon is regular but we have assumed that ourselves. If we will not consider such an assumption then there will be 11 different angles for which we will be required to form 11 linear equations to solve for the value of each angle and also there isn’t such information provided.