Question

The sum of the angles of a polygon is $3240$. How many sides does the polygon have?

Answer 1

A polygon is a plane figure that has a finite number of straight-line segments connected to form a closed polygonal chain. The bounded plane region, the bounding circuit, or the two together, maybe called a polygon. In the question, we are asked to find the number of sides in a polygon whose total measurement of interior angles is given. To solve this, we will use the concept that if a polygon has $n$ sides, then the sum of its interior angles is given by $\left( {n - 2} \right){180^ \circ }$. We will equate the given sum of the angles with this and solve the equation to get the value of $n$.

Complete step-by-step solution:
Let, there are $n$ sides in the polygon. Then we know that if a polygon has $n$ sides, then the sum of its interior angles is given as: $\left( {n - 2} \right){180^ \circ }$.
But the given sum is $= 3240$
So, equating both of them we get;
$\Rightarrow \left( {n - 2} \right)180 = 3240$
On dividing both sides by $180$, we get;
$\Rightarrow \left( {n - 2} \right) = \dfrac{{3240}}{{180}}$
On calculating we have;
$\Rightarrow n - 2 = 18$
On shifting we get;
$\Rightarrow n = 20$

So, the total number of sides in the polygon is $20$.

Note: One thing to note here is that for the sum of interior angles of a polygon we have used the formula $\left( {n - 2} \right){180^ \circ }$. This is so because, any polygon of $n$ sides can be thought of made up of $\left( {n - 2} \right)$ triangles, and we know that sum of the angles of a triangle is ${180^ \circ }$. So, the sum of angles of $\left( {n - 2} \right)$ triangles will be $\left( {n - 2} \right){180^ \circ }$.