Question

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

Answer 1

Hint: 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 \[\left( {n - 2} \right){180^ \circ }\].