Question

Let the formula relating the exterior angle and number of sides of a polygon be given as $nA = 360.$ The measure A, in degrees, of an exterior angle of a regular polygon is related to the number of sides, n, of the polygon by the formula above. If the measure of an exterior angle of a regular polygon is greater than $50$ , what is the greatest number of sides it can have?
A) $5$
B) $6$
C) $7$
D) $8$

Answer 1

Hint: The relation between number of sides of a polygon and its external angle is already given. Find the value of the external angle from this relation and substitute this in $A > 50$ which is also given. Solve this inequation to get the maximum number of sides for this external angle.

Complete step by step answer:
- Given to us, the relation between external angle measured in degrees and the number of sides of the polygon as $nA = 360$.
- From this, we can write the value of the external angle as $A = \dfrac{{360}}{n}$. It is also given that the external angle $A$ is greater than $50$ degrees. This can be written as $A > 50$.
- Now substituting the value of $A$ from the relation, we can write this as $\dfrac{{360}}{n} > 50$.
- We can rearrange this as $\dfrac{{360}}{{50}} > n$.
- This can also be written as $n < \dfrac{{360}}{{50}}$.
$n < 7.2$
- We can approximate this to the nearest number as seven.

Therefore, the maximum or greatest number of sides a polygon can have with an exterior angle greater than $50$ is $7$ sides i.e. option C.

Note: Note that the exterior angle of any polygon can never exceed or be equal to $180$ degrees. In this case, when the external angle is more than fifty degrees as given, the maximum number of sides that a polygon can have are seven.