Find the number of squares such that the sides of the square are segments in the following diagram and where the dot is inside the square.

[center][img]https://snag.gy/qXBIY4.jpg[/img][/center]
1. **Case 1: $1 \times 1$ square**
   - By inspection, there is only $1$ square that works. This is the square that contains the dot directly in its center.

2. **Case 2: $2 \times 2$ square**
   - By inspection, there are $4$ possible $2 \times 2$ squares that work. These squares are formed by choosing any $2 \times 2$ subgrid that contains the dot.

3. **Case 3: $3 \times 3$ square**
   - There are $9$ possible $3 \times 3$ squares in the grid. Since the dot is centrally located, all $9$ of these squares will contain the dot.

4. **Case 4: $4 \times 4$ square**
   - There are $4$ possible $4 \times 4$ squares in the grid. Each of these squares will contain the dot, as the dot is centrally located.

5. **Case 5: $5 \times 5$ square**
   - There is only $1$ possible $5 \times 5$ square, which is the entire grid itself. This square contains the dot.

Summing all of these cases up, we get:
\[ 1 + 4 + 9 + 4 + 1 = 19 \]

The final answer is $\boxed{19}$