Question

Each interior angle of a regular polygon is \[156\] degrees. How many sides does it have?

Answer 1

Hint: Polygon refers to a geometrical plane figure. It is made up of a finite number of straight lines which are connected to form a closed polygonal circuit. In order to find the sides of the polygon, we have to subtract the interior angle from\[{180^ \circ }\] to get the exterior angle. After this, we have to divide \[{360^ \circ }\] by the difference of the \[{180^ \circ }\] and interior angle.

Formula used:
The first formula that will be used in the question is the calculate the exterior angle
\[{\text{Exterior angle = }}{180^ \circ } - {\text{Interior angle}}\] .
The second formula will be used to calculate the number of sides of the polygon \[{\text{Number of sides of the polygon = }}\dfrac{{{{360}^ \circ }}}{{{\text{Exterior angle}}}}\] .

Complete step by step solution:
In the above question, we are given the interior angle of the polygon, which is, \[{156^ \circ }\]
Now, we have to calculate the exterior angle of the polygon. For this we will use the formula 1) \[{\text{Exterior angle = }}{180^ \circ } - {\text{Interior angle}}\] .
So, \[{\text{Exterior angle = 18}}{{\text{0}}^ \circ } - {156^ \circ }\]
\[ = {24^ \circ }\] .
Now, we will calculate the number of sides of the polygon by using the formula 2) \[{\text{Number of sides of the polygon = }}\dfrac{{{{360}^ \circ }}}{{{\text{Exterior angle}}}}\] .
We get, \[{\text{Number of sides of the polygon = }}\dfrac{{{{360}^ \circ }}}{{{{24}^ \circ }}}\]
\[ = 15\] sides.
Therefore, the number of sides of the polygon having the interior angle \[{156^ \circ }\] is \[15\] sides.

Note: In order to solve such questions involving polygons, you must remember all the formulas. In the above question, we have used two formulas: 1) \[{\text{Exterior angle = }}{180^ \circ } - {\text{Interior angle}}\] and \[{\text{Number of sides of the polygon = }}\dfrac{{{{360}^ \circ }}}{{{\text{Exterior angle}}}}\] . You cannot solve such questions without the help of these formulas. Also, with the help of these formulas you can easily solve the question in a very less amount of time.