Name of Polygons
| Polygon | | Number of Sides | Number of Dissected Triangles | Sum of Interior Angles |
| :------------ | :-: | :-------------- | :---------------------------- | :----------------------- |
| Triangle | | 3 | 1 | (1 x 180) or 180 |
| Quadrilateral | | 4 | 2 | (2 x 180) or 360 |
| Pentagon | | 5 | | |
| Hexagon | | 6 | | |
| Heptagon | | 7 | | |
| Octagon | | 8 | | |
| Dodecagon | | 12 | | |
1. Is there a connection between the number of sides and the number of dissected triangles in a polygon? What is this relation?
2. How many dissected triangles are there in an n-sided polygon? What is the sum of the interior angles of an n-sided polygon?
Answer (1)
To understand the connection between the number of sides and the number of dissected triangles in a polygon, we need to explore some basic geometric concepts.
The Connection between the Number of Sides and Dissected Triangles :
A polygon can be divided into triangles by drawing diagonals from one vertex to all other non-adjacent vertices. This means that the number of triangles is directly related to the number of sides.
For any polygon with n sides, the number of dissected triangles can be found using the formula:
\[\text{Number of triangles} = n - 2\]
This is because, when dividing the polygon from a single vertex, you do not count the sides connecting to the adjacent vertices on either side of the vertex.
Number of Dissected Triangles in an n -sided Polygon :
If you have a polygon with n sides, you can always form n − 2 triangles. This relationship holds after observing that a triangle (the simplest polygon) is a natural starting point, having 3 sides and forming 1 triangle ( 3 − 2 = 1 ).
Sum of the Interior Angles of an n -sided Polygon :
Once you know the number of triangles n − 2 , you can determine the sum of the interior angles because the sum of angles in each triangle is 180 degrees.
The formula to find the sum of the interior angles of an n -sided polygon is:
\[\text{Sum of interior angles} = (n - 2) \times 180\]
For example:
In a pentagon (5 sides): Number of triangles = 5 − 2 = 3 , so the sum of the interior angles is 3 × 180 = 540 degrees.
In a hexagon (6 sides): Number of triangles = 6 − 2 = 4 , so the sum of the interior angles is 4 × 180 = 720 degrees.
Understanding these relationships helps in solving various polygon-related problems in geometry and can be quite useful in practical applications like architecture, design, and more.
