Number of diagonal of an n-sided polygon is__________

Answer

To find the number of diagonals in an n-sided polygon, we can use a specific formula. Here’s a step-by-step solution:

### Step 1: Understand the polygon
A polygon is a closed figure with straight sides. The number of sides in the polygon is denoted by \( n \).

### Step 2: Identify the formula
The formula to calculate the number of diagonals \( D \) in an n-sided polygon is given by:
\[ D = \frac{n(n - 3)}{2} \]
where \( n \) is the number of sides of the polygon.

### Step 3: Derive the formula
To understand why this formula works, consider the following:
- Each vertex of the polygon can connect to \( n - 1 \) other vertices.
- However, two of these connections are the sides of the polygon itself (the adjacent vertices).
- Therefore, each vertex can connect to \( n - 3 \) vertices to form a diagonal.
- Since there are \( n \) vertices, the total number of connections (or diagonal endpoints) is \( n(n - 3) \).
- However, this counts each diagonal twice (once from each endpoint), so we divide by 2.

### Step 4: Example Calculation
Let’s say we want to find the number of diagonals in a hexagon (6-sided polygon).
Using the formula:
\[ D = \frac{6(6 - 3)}{2} = \frac{6 \times 3}{2} = \frac{18}{2} = 9 \]
So, a hexagon has 9 diagonals.

### Final Answer
The number of diagonals in an n-sided polygon is given by:
\[ \frac{n(n - 3)}{2} \]