A regular polygon is a polygon where all sides are equal in length and all angles are equal. The sum of the interior angles of any polygon can be calculated using the formula (n2) x 180 degrees, where n is the number of sides in the polygon.
In the case of a regular polygon with an interior angle of 165 degrees, we can determine the number of sides by dividing 360 degrees by the interior angle. Therefore, 360 degrees ÷ 165 degrees ≈ 2.182. Since a polygon cannot have a fraction of a side, we round this number down to the nearest whole number, which means our polygon has 2 sides or, more commonly expressed, is a line segment.
A polygon with an interior angle of 165 degrees is not a usual case for regular polygons, as such a large interior angle would result in less than 3 sides for the polygon, making it theoretically impossible to form a closed shape. Regular polygons typically have interior angles that are more common, such as 60, 90, 120, or 144 degrees.
For a more practical scenario, let’s consider a regular polygon with an interior angle of 165 degrees. In this case, the closest common interior angle for a regular polygon would be 150 degrees for a pentagon (5 sides). A regular pentagon has interior angles of 108 degrees, so a polygon with interior angles of 165 degrees would be irregular, given the sum of the interior angles must still equal (n2) x 180 degrees.
Properties of a Regular Polygon with an Interior Angle of 165 Degrees
Let’s delve into the properties of a regular polygon with an interior angle of 165 degrees in more detail.
1. Definition
A regular polygon with an interior angle of 165 degrees is a closed geometric shape where all sides are equal in length and all interior angles are equal to 165 degrees.
2. Unique Characteristics
 The sum of all interior angles in a regular polygon with 165 degrees is still a multiple of 180 degrees.
 The exterior angle of the polygon would be 15 degrees (180 degrees – 165 degrees).
 The number of sides can be calculated by dividing 360 by 165, yielding a nonwhole number.
3. Construction Challenges
Constructing a regular polygon with an interior angle of 165 degrees poses a challenge due to the unusual angle measurement, making it difficult to create a symmetrical shape with equal sides and angles.
4. Realworld Applications
Regular polygons with uncommon interior angles like 165 degrees are not commonly encountered in realworld applications due to their irregular nature. However, irregular polygons can be found in architecture, art, and design.
FAQs:

Can a regular polygon have an interior angle of 165 degrees?
While theoretically possible, a regular polygon with an interior angle of 165 degrees would not fit the standard definition of a regular polygon due to the unusual angle measurement. 
What is the formula to calculate the sum of interior angles in a polygon?
The formula is (n2) x 180 degrees, where n represents the number of sides in the polygon. 
How many sides would a regular polygon with an interior angle of 165 degrees have?
By dividing 360 degrees by 165 degrees, the result is approximately 2.182 sides, which is not feasible for a polygon. 
What is the closest common interior angle for a regular polygon with 165 degrees?
The closest common interior angle would be 150 degrees, often seen in a regular pentagon with 5 sides. 
What practical applications involve irregular polygons with unusual interior angles?
Irregular polygons with uncommon interior angles can be found in artistic designs, architectural layouts, and specialized geometric constructions where symmetry is not a primary concern.