Question

If the sum of the measures of the interior angles of a polygon is $ {5400^ \circ } $ , how many sides does it have?

Answer 1

Hint: We know that a polygon can be defined as any geometric shape that is enclosed by a number of straight sides. Interior angle is the angle created by these straight lines inside the polygon. To solve this question, we need to use the relation between the sum of the measures of all the interior angles and the total number of sides of the polygon.
 Formula used:
 $ S = \left( {n - 2} \right) \times {180^ \circ } $ , Where $ S $ is the sum of the measures of the interior angles of a polygon and $ n $ is the number of sides the polygon has.

Complete step by step solution:
We are given that the sum of the measures of the interior angles of a polygon is $ {5400^ \circ } $ . This means that the value of $ S $ is $ {5400^ \circ } $ .
We have to find the number of sides of this polygon which is $ n $ .
Now, we will apply the formula $ S = \left( {n - 2} \right) \times {180^ \circ } $ and put the value of $ S $ in it. By doing this, we will have a linear equation with variable $ n $ by solving which we can get the number of the sides of the given polygon.
 $
  S = \left( {n - 2} \right) \times {180^ \circ } \\
   \Rightarrow {5400^ \circ } = \left( {n - 2} \right) \times {180^ \circ } \\
   \Rightarrow \left( {n - 2} \right) = \dfrac{{{{5400}^ \circ }}}{{{{180}^ \circ }}} \\
   \Rightarrow n - 2 = 30 \\
   \Rightarrow n = 32 \;
  $
Thus, the polygon has 32 sides.
So, the correct answer is “32 sides”.

Note: Here, we have determined the number of sides of the polygon. Now, we have the sum of measures of all the interior angles and the number of sides. By using these both, we can calculate the measure of a single interior angle in case of a regular polygon as:
Interior angle $ = \dfrac{S}{n} = \dfrac{{{{5400}^ \circ }}}{{32}} = {168.75^ \circ } $ .