How Many Diagonals Are in an N-gon?

In geometry, a diagonal is a line segment that connects two non-adjacent vertices of a polygon. An n-gon is a polygon with n sides. For example, a triangle is a 3-gon, a quadrilateral is a 4-gon, a pentagon is a 5-gon, and so on.

To find the number of diagonals in an n-gon, we can use the following formula:

**Number of diagonals = n(n-3)/2**

Where 'n' is the number of sides of the polygon.

Derivation of the Formula

Let's understand how this formula is derived. Consider an n-gon. From each vertex, we can draw (n-3) diagonals. For example, in a pentagon (n=5), from each vertex, we can draw 2 (5-3) diagonals.

However, if we simply multiply n by (n-3), we are counting each diagonal twice (once for each endpoint). Therefore, we divide the result by 2 to get the actual number of diagonals.

Example

Let's find the number of diagonals in a hexagon (n=6).

Number of diagonals = 6(6-3)/2 = 9

Therefore, a hexagon has 9 diagonals.

Table of Diagonals

Polygon	Number of Sides (n)	Number of Diagonals
Triangle	3	0
Quadrilateral	4	2
Pentagon	5	5
Hexagon	6	9
Heptagon	7	14
Octagon	8	20

Applications

Understanding the concept of diagonals is crucial in various geometric applications, including:

• Calculating the area and perimeter of polygons.
• Analyzing the properties of regular polygons.
• Solving problems related to symmetry and tessellations.

Conclusion

The formula for calculating the number of diagonals in an n-gon provides a simple and effective method to determine this geometric property. This knowledge is valuable for various geometric applications and helps in understanding the relationships between sides, vertices, and diagonals in polygons.