Understanding the Interior Angle of a 20-Sided Polygon (Icosagon)

Imagine a polygon with 20 sides, also known as an icosagon. Want to know the size of each of its interior angles? This article will provide a clear and concise explanation of how to calculate the interior angle of a 20-sided regular polygon, along with useful formulas and practical applications.

Calculating the Interior Angle of a Regular Polygon

To find the interior angle of a regular polygon with n sides, use the following formula:

[text{Interior Angle} frac{(n - 2) times 180^circ}{n}]

This formula can be simplified for a 20-sided polygon, where n 20:

[text{Interior Angle} frac{(20 - 2) times 180^circ}{20} frac{18 times 180^circ}{20} frac{3240^circ}{20} 162^circ]

Thus, the interior angle of a regular 20-sided polygon, or icosagon, is 162°.

SUM OF EXTERIOR ANGLES OF A POLYGON

It’s important to understand that the sum of the exterior angles of any polygon is always 360°, regardless of the number of sides. This principle can be used to find individual exterior and interior angles of a regular polygon. For a regular icosagon:

[text{Each exterior angle} frac{360^circ}{20} 18^circ]

The interior angle is then the supplementary angle to the exterior angle:

[text{Interior angle} 180^circ - 18^circ 162^circ]

Interior and Exterior Angles of Regular Polygons

In a regular polygon, all sides and angles are equal. For a regular icosagon with 20 sides:

[text{Exterior angle} frac{360^circ}{20} 18^circ]

The interior angle can be calculated as:

[text{Interior angle} 180^circ - 18^circ 162^circ]

General Formula for Any Regular Polygon

For any regular polygon with n sides, the interior angle can be calculated using the general formula:

[text{Interior angle} 180^circ - frac{360^circ}{n}]

Substituting n 20 into the formula:

[text{Interior angle} 180^circ - frac{360^circ}{20} 180^circ - 18^circ 162^circ]

Or, using the shorthand version of the formula:

[text{Interior angle} frac{(20 - 2) times 180^circ}{20} frac{18 times 180^circ}{20} 162^circ]

Conclusion

The interior angle of a regular 20-sided polygon, or icosagon, is 162°. This value is consistent with the calculations derived from different formulas and methods. Understanding the properties and angles of regular polygons is crucial in various fields, including architecture, engineering, and mathematics.

By now, you should have a clear understanding of how to calculate the interior angle of a 20-sided polygon and how it relates to the sum of the exterior angles. If you need further information or have any other questions about the properties of polygons, feel free to explore more resources or ask your queries in the comments below.