No, it is not possible to have a regular polygon whose interior angle is exactly 110 degrees.
Understanding Regular Polygons
A regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Examples include a square (four equal sides and 90-degree angles) or a regular pentagon (five equal sides and 108-degree angles).
For any polygon, there's a fundamental relationship between its interior angles and its exterior angles. An interior angle is an angle inside the polygon at a vertex, while an exterior angle is formed by one side of the polygon and the extension of an adjacent side. The sum of an interior angle and its corresponding exterior angle at any vertex is always 180 degrees.
The Calculation for 110 Degrees
To determine if a regular polygon can have an interior angle of 110 degrees, we follow a specific mathematical process:
Step-by-Step Analysis
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Calculate the Exterior Angle:
The sum of an interior angle and its corresponding exterior angle is 180 degrees.- Exterior Angle = 180° - Interior Angle
- Exterior Angle = 180° - 110° = 70°
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Determine the Number of Sides (n):
For any regular polygon, the sum of all its exterior angles is always 360 degrees. Therefore, each exterior angle of a regular polygon with 'n' sides is 360° divided by 'n'. We can use this to find 'n':- n = 360° / Exterior Angle
- n = 360° / 70°
- n = 5.14285714286...
Why 110 Degrees Is Not Possible
The number of sides of a polygon, represented by 'n', must always be a natural number (a positive whole number, such as 3, 4, 5, 6, and so on). Since our calculation for 'n' yielded approximately 5.14, which is not a whole number, it is geometrically impossible to construct a regular polygon with an interior angle of 110 degrees. You cannot have a fraction of a side in a polygon.
Properties of Regular Polygons
For a regular polygon to exist, both its interior and exterior angles must result in a whole number of sides when calculated. This property ensures the polygon can close perfectly without gaps or overlaps.
Here are some examples of regular polygons with their interior and exterior angles, demonstrating how they correspond to a whole number of sides:
| Number of Sides (n) | Name | Interior Angle | Exterior Angle |
|---|---|---|---|
| 3 | Equilateral Triangle | 60° | 120° |
| 4 | Square | 90° | 90° |
| 5 | Regular Pentagon | 108° | 72° |
| 6 | Regular Hexagon | 120° | 60° |
| 8 | Regular Octagon | 135° | 45° |
As you can see from the table, an interior angle of 110 degrees falls between the angles of a regular pentagon (108°) and a regular hexagon (120°), indicating that it would require a number of sides between 5 and 6, which is not possible for a polygon.
For more detailed information on polygons and their properties, you can explore resources like Math is Fun.
Conclusion
The impossibility of having a regular polygon with an interior angle of 110 degrees stems directly from the mathematical requirement that a polygon must have a whole, natural number of sides. Since the calculation results in a fractional number of sides, such a polygon cannot exist.
