The exact exterior angle sum of a decagon is 360 degrees.
Understanding the properties of polygons, especially their angles, is fundamental in geometry. A decagon, a polygon with ten sides, follows a universal rule regarding its exterior angles.
Understanding Exterior Angles
An exterior angle of a polygon is formed by extending one of its sides and the adjacent side. When discussing convex polygons, the sum of these exterior angles always remains constant, regardless of the number of sides the polygon has.
What is a Decagon?
A decagon is a two-dimensional polygon characterized by:
- 10 sides
- 10 vertices (corners)
- 10 interior angles
- 10 exterior angles
The term "deca" means ten, and "gon" means angle. To learn more about decagons, you can visit Wikipedia's Decagon page.
Calculating the Exterior Angle Sum of a Decagon
While the sum of exterior angles for any convex polygon is a fixed value, we can demonstrate this specifically for a decagon through its properties.
Steps to Calculate for a Regular Decagon
For a regular decagon (where all sides and all angles are equal):
-
Determine the sum of interior angles: The formula for the sum of interior angles of any polygon with n sides is
(n - 2) × 180°.- For a decagon (n=10):
(10 - 2) × 180° = 8 × 180° = 1440°.
- For a decagon (n=10):
-
Calculate each interior angle: For a regular decagon, divide the total interior angle sum by the number of sides.
- Each interior angle =
1440° / 10 = 144°.
- Each interior angle =
-
Find each exterior angle: Each exterior angle and its corresponding interior angle form a linear pair, meaning they add up to 180°.
- One exterior angle =
180° - Interior Angle = 180° - 144° = 36°.
- One exterior angle =
-
Sum the exterior angles: Since a decagon has 10 exterior angles, and each one is 36° in a regular decagon:
- Sum of exterior angles =
10 × 36° = 360°.
- Sum of exterior angles =
This calculation confirms that the sum of the exterior angles of a decagon is 360°.
The Universal Rule for All Convex Polygons
It's important to note that this sum of 360° is not unique to decagons. The sum of the exterior angles of any convex polygon, regardless of the number of sides, is always 360 degrees. This is a fundamental principle in geometry.
For example:
- A triangle (3 sides) has an exterior angle sum of 360°.
- A quadrilateral (4 sides) has an exterior angle sum of 360°.
- A hexagon (6 sides) has an exterior angle sum of 360°.
This principle can be visualized by imagining "walking" around the perimeter of the polygon, turning at each vertex. By the time you return to your starting point and orientation, you will have completed a full 360-degree rotation.
Decagon Angle Summary
To provide a clear overview, here's a summary of angles related to a regular decagon:
| Angle Type | Formula / Value for Regular Decagon |
|---|---|
| Number of Sides (n) | 10 |
| Sum of Interior Angles | (n - 2) × 180° = 1440° |
| Each Interior Angle | 1440° / 10 = 144° |
| Each Exterior Angle | 180° - 144° = 36° |
| Sum of Exterior Angles | 360° |
Understanding this consistent property of exterior angles simplifies many geometric problems and offers a deep insight into polygon characteristics.
