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Find the number of sides of a regular polygon if each exterior angle is equal to one-third of its adjacent interior angle.

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Last updated date: 13th Jun 2024
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Answer
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Hint: Here we use the concept that both exterior angle and adjacent interior angle are supplementary to each other. So assuming the angles as two different variables we form an equation. Using the relation given in the statement form another equation. Solve both the equations to find values of the angles. Then use formula for exterior angle to find number of sides of polygon.
* If there are n sides of a polygon, then the exterior angle is given by \[\dfrac{{{{360}^ \circ }}}{n}\].

Complete step-by-step answer:
In any polygon, the exterior angle is the angle which lies outside the polygon but is on a line which is extended from the polygon. Also, the adjacent interior angle means the interior angle which is adjacent (side by side) to the exterior angle. So, we can say both adjacent interior angle and exterior angle lie on the same straight line which means they are supplementary.
Let us assume the exterior angle as \[x\] and the adjacent interior angle as \[y\].
Since, \[x\] and \[y\] are supplementary.
\[ \Rightarrow x + y = {180^ \circ }\] … (1)
Now we are given in the statement that each exterior angle is equal to one-third of its adjacent interior angle. So every \[x\] is one third of\[y\].
\[ \Rightarrow x = \dfrac{1}{3}y\]
Multiplying both sides by 3 we can write
\[ \Rightarrow 3x = y\] … (2)
Substitute the value from equation (2) in equation (1).
\[
   \Rightarrow x + 3x = {180^ \circ } \\
   \Rightarrow 4x = {180^ \circ } \\
 \]
Divide both sides of the equation by 4
\[
   \Rightarrow \dfrac{{4x}}{4} = \dfrac{{{{180}^ \circ }}}{4} \\
   \Rightarrow x = {45^ \circ } \\
 \]
Substituting the value in equation (2) we get
\[
   \Rightarrow 3({45^ \circ }) = y \\
   \Rightarrow y = {135^ \circ } \\
 \]
Now we use the formula of exterior angle which is given by number of sides of a polygon i.e. exterior angle \[ = \dfrac{{{{360}^ \circ }}}{n}\]
We know the value of the exterior angle is \[{45^ \circ }\].
\[ \Rightarrow {45^ \circ } = \dfrac{{{{360}^ \circ }}}{n}\]
Cross multiply the value of \[n\] and \[{45^ \circ }\]
\[ \Rightarrow n = \dfrac{{{{360}^ \circ }}}{{{{45}^ \circ }}} = 8\]
So, the number of sides of the polygon are 8.

Note: Students can mostly get confused with the term one-third and might make wrong equation of the statement, keep in mind one third of other angle means we multiply \[\dfrac{1}{3}\]to the angle to which we are relating and equate it with the angle about which we are stating the statement.