Question

The sum of the interior angles of a polygon is four times the sum of its exterior angles. Find the number of sides in the polygon.
A. 10
B. 12
C. 8
D. 7

Answer 1

Hint: First of all, consider the required number of sides of the polygon as a variable. The sum of exterior angles of a polygon is always equal to \[{360^\circ}\]. The sum of interior angles of a polygon is equal to \[\left( {n - 2} \right)\pi \]. So, use this concept to reach the solution of the given problem.

Complete step-by-step answer:
Given the sum of the interior angles of a polygon is four times the sum of its exterior angles.
We know that the sum of exterior angles of a polygon is always equal to \[{360^\circ}\].
So, sum of interior angles of the polygon \[ = 4 \times {360^\circ} = {1440^\circ}\]
Let \[n\] be the number of sides of a polygon.
We know that sum of interior angles of a polygon \[ = \left( {n - 2} \right)\pi \]
By using the above data, we have
\[
  \Rightarrow \left( {n - 2} \right)\pi = {1440^\circ} \\
  \Rightarrow \left( {n - 2} \right){180^\circ} = {1440^\circ} \\
  \Rightarrow n - 2 = \dfrac{{{{1440}^\circ}}}{{{{180}^\circ}}} \\
  \Rightarrow n - 2 = 8 \\
  \therefore n = 8 + 2 = 10 \\
\]
Hence, the number of sides of the required polygon is 10.
Thus, the correct answer is A. 10

Note: In geometry, a polygon is a plane figure that is described by a finite number of straight-line segments connected to form a closed polygonal chain or polygonal circuit. A polygon of 10 sides is called a decagon.