Determining the Interior Angle of a Regular 20-Sided Polygon
Understanding the interior angles of polygons is an important concept in geometry, especially for regular polygons, which have all sides and angles equal. This article explores how to calculate the interior angle of a regular 20-sided polygon, or icosagon, using both the interior angle formula and the properties of exterior angles.
Formula for Interior Angle of a Regular Polygon
The formula to calculate the interior angle of a regular polygon is:
Interior Angle (n - 2) × 180° / n
Where n is the number of sides. For a regular polygon with 20 sides (a 20-sided polygon or an icosagon), we can substitute n 20 into the formula:
Interior Angle (20 - 2) × 180° / 20
Simplifying this expression:
Interior Angle 18 × 180° / 20Interior Angle 3240° / 20Interior Angle 162°
Therefore, each interior angle of a regular 20-sided polygon is 162 degrees.
Using Properties of Exterior Angles
The sum of exterior angles of any polygon is always 360 degrees. For a regular polygon, all the exterior angles are equal. To find the measure of each exterior angle of a 20-sided polygon, we divide 360 degrees by the number of sides:
Exterior Angle 360° / 20 18°
Since the interior and exterior angles are supplementary (they add up to 180 degrees), we can find the interior angle by subtracting the exterior angle from 180 degrees:
Interior Angle 180° - 18° 162°
Calculations for a Polygon with 20 Sides
For an n-sided polygon, the formula to calculate the interior angle is a general formula applicable for any regular polygon:
Interior Angle (n - 2) × 180° / n
Substituting n 20:
Interior Angle (20 - 2) × 180° / 20
Further simplification:
Interior Angle 18 × 180° / 20Interior Angle 3240° / 20Interior Angle 162°
This confirms that the interior angle of a regular 20-sided polygon is 162 degrees.
Conclusion
In conclusion, the interior angle of a regular 20-sided polygon (icosagon) is 162 degrees. This can be calculated using the general formula for the interior angle of a regular polygon, or by leveraging the properties of exterior angles. Understanding these relationships is fundamental for solving geometric problems and enhances your comprehension of polygonal structures.