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Question

The number of diagonals in a polygon with n sides is

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1. **Total Segments**: A polygon with $n$ vertices allows $\binom{n}{2}$ total ways to join any two vertices.
   - Total Segments $= \frac{n(n-1)}{2}$.
2. **Filter Segments**: These segments consist of either **sides** or **diagonals**.
3. **Subtract Sides**: There are $n$ sides in an $n$-sided polygon.
   - Diagonals $= \frac{n(n-1)}{2} - n$.
4. **Simplification**:
   - $\frac{n^2 - n - 2n}{2} = \frac{n^2 - 3n}{2} = \frac{n(n-3)}{2}$.
5. **Example**: A square ($n=4$) has $\frac{4(4-3)}{2} = 2$ diagonals.

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