Find the area of the hexagon whose consecutive vertices are `(5, 0), (4, 2), (1, 3), (-2, 2), (-3, -1) and (0, -4)`

Answer

To find the area of the hexagon with the given vertices, we can use the formula for the area of a polygon based on its vertex coordinates. The formula is:

\[ \text{Area} = \frac{1}{2} \left| \sum_{i=1}^{n} (x_i y_{i+1} - y_i x_{i+1}) \right| \]

where \( (x_{n+1}, y_{n+1}) \) is the same as \( (x_1, y_1) \) to close the polygon.

### Step-by-Step Solution:

1. **List the Coordinates**:
The vertices of the hexagon are given as:
- \( A(5, 0) \)
- \( B(4, 2) \)
- \( C(1, 3) \)
- \( D(-2, 2) \)
- \( E(-3, -1) \)
- \( F(0, -4) \)

2. **Set Up the Formula**:
We will apply the area formula for the vertices \( A, B, C, D, E, F \). We will also repeat the first vertex at the end:
- \( (5, 0), (4, 2), (1, 3), (-2, 2), (-3, -1), (0, -4), (5, 0) \)

3. **Calculate the Determinants**:
Using the formula, we calculate:
\[ \text{Area} = \frac{1}{2} \left| (5 \cdot 2 + 4 \cdot 3 + 1 \cdot 2 + (-2) \cdot (-1) + (-3) \cdot (-4) + 0 \cdot 0) - (0 \cdot 4 + 2 \cdot 1 + 3 \cdot (-2) + 2 \cdot (-3) + (-1) \cdot 0 + (-4) \cdot 5) \right| \]

4. **Perform the Multiplications**:
- First part:
- \( 5 \cdot 2 = 10 \)
- \( 4 \cdot 3 = 12 \)
- \( 1 \cdot 2 = 2 \)
- \( -2 \cdot -1 = 2 \)
- \( -3 \cdot -4 = 12 \)
- \( 0 \cdot 0 = 0 \)
- Sum of first part = \( 10 + 12 + 2 + 2 + 12 + 0 = 38 \)

- Second part:
- \( 0 \cdot 4 = 0 \)
- \( 2 \cdot 1 = 2 \)
- \( 3 \cdot -2 = -6 \)
- \( 2 \cdot -3 = -6 \)
- \( -1 \cdot 0 = 0 \)
- \( -4 \cdot 5 = -20 \)
- Sum of second part = \( 0 + 2 - 6 - 6 + 0 - 20 = -30 \)

5. **Combine the Results**:
\[ \text{Area} = \frac{1}{2} \left| 38 - (-30) \right| = \frac{1}{2} \left| 38 + 30 \right| = \frac{1}{2} \left| 68 \right| = \frac{68}{2} = 34 \]

6. **Final Result**:
The area of the hexagon is \( 34 \) square units.