Question

The sum of the measures of the interior angles of a convex polygon is $540$. How do you classify the polygon by the number of sides?

Answer 1

Given a measure of an angle. We have to determine the number of sides using the measure of an angle. First, we will substitute the value of the angle into the formula of the sum of interior angles of a convex polygon. Then, simplify the expression and solve for the number of sides. Then according to the number of sides, we will classify the polygon.

Formula used:
The sum of the interior angles of convex polygon is given by:
$Sum = \left( {n - 2} \right) \times 180^\circ$
Where $n$ is the number of sides of the polygon.

Complete step-by-step answer:
We are given the sum of the interior angle. First, we will apply the formula for the sum of the interior angles of the convex polygon by substituting $Sum = 540^\circ$ into the formula.
$\Rightarrow \left( {n - 2} \right) \times 180^\circ = 540^\circ$
Divide both sides by $180^\circ$, we get:
$\Rightarrow \dfrac{{\left( {n - 2} \right) \times 180^\circ }}{{180^\circ }} = \dfrac{{540^\circ }}{{180^\circ }}$
$\Rightarrow n - 2 = 3$
Now, add $2$ to both sides of the expression.
$\Rightarrow n - 2 + 2 = 3 + 2$
$\Rightarrow n = 5$

Hence, the number of sides is five which means the polygon is classified as a pentagon.

Additional Information: The interior angles of a particular polygon are the angles that are formed inside the two adjacent sides. There can be two types of a polygon. One is the polygon in which each interior angle is of the same measure which is known as a regular polygon. Another type of polygon is an irregular polygon in which the measure of each angle is different.

Note:
In such types of questions the students mainly don't get an approach on how to solve it. In such types of questions students are mainly confused while applying the formula for the sum of interior angles of the convex polygon.