Chelsea drew a 16-sided polygon.
What is the sum of the interior angles of the 16-sided polygon?
The correct answer and explanation is :
To determine the sum of the interior angles of a 16-sided polygon, we can apply a standard formula used for any polygon with ‘n’ sides:
Sum of Interior Angles = (n – 2) × 180°
In this case, the polygon has 16 sides, so:
Sum of Interior Angles = (16 – 2) × 180° = 14 × 180° = 2,520°
This means that the sum of all interior angles of a 16-sided polygon is 2,520 degrees.
Explanation:
A polygon is a two-dimensional geometric figure with straight sides that are fully closed. The number of sides (n) significantly influences the sum of its interior angles. To understand why the formula (n – 2) × 180° works, consider the following:
- Triangulation Method: Any polygon can be divided into triangles by drawing diagonals from one vertex to all other non-adjacent vertices. Each triangle has interior angles summing to 180°. The number of triangles formed in an n-sided polygon is (n – 2). Therefore, the total sum of interior angles is (n – 2) × 180°.
- Example with a Triangle (n = 3): Dividing a triangle into (3 – 2) = 1 triangle gives 1 × 180° = 180°, which matches the sum of interior angles of a triangle.
- Example with a Quadrilateral (n = 4): Dividing a quadrilateral into (4 – 2) = 2 triangles gives 2 × 180° = 360°, which matches the sum of interior angles of a quadrilateral.
Applying this reasoning to a 16-sided polygon:
- Number of Triangles: (16 – 2) = 14 triangles.
- Sum of Interior Angles: 14 × 180° = 2,520°.
This consistent method allows us to calculate the sum of interior angles for any polygon, regardless of whether it’s regular (all sides and angles equal) or irregular.