Question

The sum of interior angles and the sum of exterior angles of a polygon are in the ratio \[9:2\]. Find the number of sides in the polygon.

Answer 1

Hint: When sides are given, we already know the method for calculating the sum of interior angles of a polygon, and we also know that the sum of exterior angles of every polygon remains the same, i.e., \[360^\circ \]. Now we'll just arrange the equations in a way where we can compare them and figure out how many sides they have.

Complete step-by-step solution:
Let’s take the sum of interior angles as \[{S_i}\] and the sum of exterior angles are \[{S_e}\].
It is given that,
\[\dfrac{{{S_i}}}{{{S_e}}} = \dfrac{9}{2}\] \[ \to \left( 1 \right)\]
Now, we know the sum of interior angle is,
\[{S_i} = 180^\circ (n - 2)\]
And the sum of exterior angles is always \[360^\circ \] i.e.,
\[{S_e} = 360^\circ \]
Therefore, we can write this as,
\[\dfrac{{{S_i}}}{{{S_e}}} = \dfrac{{180^\circ (n - 2)}}{{360^\circ }}\] \[ \to \left( 2 \right)\]
By combining both the equation (1) and (2), we get
\[\dfrac{{{S_i}}}{{{S_e}}} = \dfrac{{180^\circ (n - 2)}}{{360^\circ }} = \dfrac{9}{2}\]
Now by cross multiplication method, we get
\[ \Rightarrow 2 \times 180^\circ (n - 2) = 9 \times 360^\circ \]
Solving this further, we will get
\[ \Rightarrow 2(n - 2) = 9 \times 2\]
\[ \Rightarrow (n - 2) = 9\]
Now, taking 2 to another side, we get
\[ \Rightarrow n = 9 + 2\]
\[ \Rightarrow n = 11\]
Hence, the total number of sides in this polygon is 11.
Now we know the sides of this polygon and so we can also determine the sum of interior angles by using the formula. We know,
\[{S_i} = 180^\circ (n - 2)\]
Putting the value of n as 11, we get
\[ \Rightarrow {S_i} = 180^\circ (11 - 2)\]
\[ \Rightarrow {S_i} = 180^\circ (9)\]
\[ \Rightarrow {S_i} = 1620^\circ \]
Hence, the sum of interior angles is \[1620^\circ \].

Note: Remember that we were not given the sides of the polygon or the sum of internal angles in this question, so we had to figure it out using the ratios. However, in other questions, you will be given either sides or the sum of interior angles to get the solution. There is no other way of solving this problem as we have the formula of sum of interior angles and exterior angles.