The sum of the interior angles of an undecagon is 1620 degrees.
An undecagon is a polygon characterized by having eleven sides. Determining the sum of its interior angles is a core concept in geometry, relying on a consistent formula applicable to all convex polygons.
Calculating the Sum of Interior Angles of an Undecagon
The total measure of the interior angles of any convex polygon can be found using a specific formula that connects the number of sides to the sum of its internal angles.
- The Formula: The sum of the interior angles (S) of a polygon with n sides is calculated as:
S = (n - 2) × 180°
To apply this formula for an undecagon:
- Identify the number of sides (n): An undecagon has 11 sides, so n = 11.
- Substitute the value of 'n' into the formula: S = (11 - 2) × 180°
- Perform the subtraction: S = 9 × 180°
- Carry out the multiplication: S = 1620°
Therefore, the sum of the interior angles of an undecagon is 1620 degrees.
Understanding Polygons and Their Angle Sums
This universal formula applies uniformly across various polygons, demonstrating a fundamental principle in geometry. The table below illustrates how the sum of interior angles varies with the number of sides, including the undecagon:
| Number of Sides (n) | Name of the Polygon | Sum of the Interior Angles (in degrees) |
|---|---|---|
| 11 | Undecagon | 180 x (11 - 2) = 1620 |
| 12 | Dodecagon | 180 x (12 - 2) = 1800 |
| 13 | Tridecagon | 180 x (13 - 2) = 1980 |
| 14 | Tetradecagon | 180 x (14 - 2) = 2160 |
For a deeper understanding of how to calculate interior angles and other polygon properties, you can refer to educational resources such as this lesson on Interior Angles of a Polygon.
Key Insights from the Formula
- The formula (n - 2) × 180° is applicable to all convex polygons, regardless of their specific shape (regular or irregular), as long as they are convex.
- The term
(n - 2)represents the number of non-overlapping triangles that can be formed by drawing diagonals from a single vertex within the polygon. Since each triangle contains 180 degrees, multiplying by 180 gives the total sum for the polygon. - As the number of sides of a polygon increases, the sum of its interior angles also increases proportionally.
