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How many sides does a regular polygon have if it has:
a) An exterior angle of \[{5^ \circ }\]
b) An interior angle of \[{178^ \circ }\]

Last updated date: 23rd May 2024
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Hint: Here in the given question to solve we need to know about the regular polygon and its properties. When exterior angle is given, use \[\dfrac{{{{360}^ \circ }}}{n}\], where n is the number of sides. The exterior angle and interior are measured from the same line thus they sum up to \[{180^ \circ }\]. In this question we are asked to find the sides so we need to alter the formula, by interchanging the unknown that is side with the angle.

Complete step-by-step solution:
A regular polygon is a 2 dimensional plane figure whose sides are formed only using line segments; these sides will be congruent to each other. In a regular polygon all the exterior angles will be equal and all the interior angles will be equal.
Exterior angles are formed outside the polygon by extending one side of the polygon and measuring the angle between that extension and adjacent sides. In a regular polygon, all the exterior angles sum up to \[{360^ \circ }\].
a) Given exterior angle is \[{5^ \circ }\]
Exterior angle =\[\dfrac{{{{360}^ \circ }}}{n}\], n is number of sides
On rearranging, we have
\[ \Rightarrow \] \[n = \dfrac{{{{360}^ \circ }}}{{Exterior\,angle}}\]----------(1)
On substituting exterior angle = \[{5^ \circ }\]
\[ \Rightarrow n = \dfrac{{{{360}^ \circ }}}{{{5^ \circ }}}\]
On simplifying we get
\[ \Rightarrow n = 72\]
The number of sides of regular polygon with exterior angle \[{5^ \circ }\] is 72
The interior angles are formed inside the polygon between the adjacent sides and are equal to each other. The number of sides and the number of interior angles will always be equal.
b) Given an interior angle of a regular polygon is \[{178^ \circ }\]
\[ \Rightarrow \]interior angle + exterior angle = \[{180^ \circ }\]
\[ \Rightarrow \]exterior angle = \[{180^ \circ }\]- interior angle
\[ \Rightarrow \]exterior angle = \[{180^ \circ }\]- \[{178^ \circ }\]
\[ \Rightarrow \]exterior angle = \[{2^ \circ }\]
Now substituting exterior angle = \[{2^ \circ }\] in equation (1)
\[ \Rightarrow n = \dfrac{{{{360}^ \circ }}}{{{2^ \circ }}}\]
\[ \Rightarrow n = 180\]
The number of sides of a regular polygon with interior angle \[{178^ \circ }\] is 180.

Note: The value of an interior angle of a regular polygon can also be calculated if the number of sides are known by using interior angle \[ = \dfrac{{(n - 2) \times {{180}^ \circ }}}{n}\], where n is the number of sides. Using these exterior and interior angle formulas we find the sides of any given polygon when asked specifically.