How Many Sides Does a Regular Polygon Have if Its Interior Angle Measures 165 Degrees?
A regular polygon, specifically one where each interior angle measures 165 degrees, possesses 24 sides. This precise number is derived from fundamental geometric principles relating a polygon's interior and exterior angles.
Understanding Regular Polygons
A regular polygon is a closed two-dimensional shape with all its sides of equal length and all its interior angles of equal measure. Common examples include equilateral triangles, squares, and regular hexagons. The number of sides largely determines the shape and angular properties of the polygon.
The Relationship Between Interior and Exterior Angles
At any given vertex of a polygon, the interior angle and its corresponding exterior angle form a linear pair, meaning they add up to 180 degrees. This relationship is key to determining the number of sides when an angle measure is known.
Here's the calculation breakdown for a regular polygon with a 165-degree interior angle:
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Calculate the Exterior Angle:
- Since the interior and exterior angles at each vertex sum to 180 degrees, if the interior angle is 165 degrees, the exterior angle must be:
- 180° - 165° = 15°
- Since the interior and exterior angles at each vertex sum to 180 degrees, if the interior angle is 165 degrees, the exterior angle must be:
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Sum of Exterior Angles:
- A fundamental property of any convex polygon, regular or irregular, is that the sum of all its exterior angles is consistently 360 degrees.
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Determine the Number of Sides:
- For a regular polygon, all exterior angles are equal. Therefore, to find the number of sides (n), you divide the total sum of exterior angles by the measure of a single exterior angle:
- n = Sum of Exterior Angles / Measure of One Exterior Angle
- n = 360° / 15°
- n = 24 sides
- For a regular polygon, all exterior angles are equal. Therefore, to find the number of sides (n), you divide the total sum of exterior angles by the measure of a single exterior angle:
Thus, a regular polygon with an interior angle of 165 degrees will have 24 sides.
Why 24 Sides for a 165-Degree Interior Angle?
The calculation explicitly demonstrates that the unique relationship between a regular polygon's interior angles, exterior angles, and the fixed sum of exterior angles (360°) dictates its number of sides. When each "turn" or exterior angle is 15 degrees, it takes 24 such turns to complete a full 360-degree rotation around the polygon, hence 24 sides.
Key Geometric Formulas for Polygons
Understanding these formulas helps in analyzing various polygon properties:
| Concept | Formula/Description |
|---|---|
| Interior Angle + Exterior Angle (at each vertex) | 180° |
| Sum of Exterior Angles (for any convex polygon) | 360° |
| Sum of Interior Angles (where n is the number of sides) | (n - 2) * 180° |
| Measure of One Exterior Angle (Regular Polygon) | 360° / n |
| Measure of One Interior Angle (Regular Polygon) | (n - 2) * 180° / n or 180° - (360° / n) |
Practical Insights and Examples
- As the number of sides increases, the measure of each interior angle of a regular polygon approaches 180 degrees, and the polygon itself begins to visually resemble a circle.
- A regular triangle (equilateral triangle) has 3 sides and an interior angle of 60°.
- A square has 4 sides and an interior angle of 90°.
- A regular pentagon has 5 sides and an interior angle of 108°.
- A regular hexagon has 6 sides and an interior angle of 120°.
The principle remains constant: the number of sides is always determined by how many times a polygon's exterior angle fits into 360 degrees.
For more information on regular polygons and their properties, you can explore resources like Math is Fun: Regular Polygons or Khan Academy: Polygon Angle Sums.
In conclusion, for a regular polygon with an interior angle of 165 degrees, the exact number of sides is 24.
