Question

A regular polygon has interior angles of \[{{168}^{\circ }}\]. How many sides does it have?

Answer 1

Hint: In this problem, we have to find the number of sides of a regular polygon which has an interior angle of \[{{168}^{\circ }}\]. We know that if we find the size of an exterior angle for the given regular polygon whose interior angle is \[{{168}^{\circ }}\], we can find the number of sides. We can subtract \[{{180}^{\circ }}\] and \[{{168}^{\circ }}\] to get the exterior angle and we can substitute it in the formula to get the number of sides.

Complete step by step solution:
We know that the given regular polygon has interior angles of \[{{168}^{\circ }}\].
We know that if we find the size of an exterior angle for the given regular polygon whose interior angle is \[{{168}^{\circ }}\], we can find the number of sides.
\[\Rightarrow {{360}^{\circ }}\div \theta =n\] …….. (1)
Where, n is the number of sides.
We can find the exterior angle by subtracting \[{{180}^{\circ }}\] with the given interior angle, we get
\[\Rightarrow {{180}^{\circ }}-{{168}^{\circ }}={{12}^{\circ }}\]
We can now substitute the above value of \[\theta \] in (1), we get
\[\Rightarrow {{360}^{\circ }}\div {{12}^{\circ }}=30\]
Therefore, there are 30 sides in the given regular polygon whose interior angle is \[{{168}^{\circ }}\].

Note: Students make mistakes while finding the exterior angle from the interior angle by subtracting the given angle with \[{{180}^{\circ }}\] which is substituted to the formula for the number of sides. We should also know that if we know the number of sides, we can find the size of the exterior angle and vice versa.