The given figure is a hexagon, and we are tasked with finding its area. Based on the grid and the coordinates of the vertices of the hexagon, we can calculate the area using geometric methods such as the Shoelace Theorem (or Gauss's area formula for polygons).

Step 1: Identify the Coordinates of the Vertices

From the image, the vertices of the hexagon appear to be:

• (-4, 0)
• (-2, 3)
• (2, 3)
• (4, 0)
• (2, -3)
• (-2, -3)

Step 2: Apply the Shoelace Theorem

The Shoelace Theorem states that the area of a polygon can be calculated as:

$\text{Area} = \frac{1}{2} \left| \sum_{i=1}^{n-1} (x_i y_{i+1} - y_i x_{i+1}) + (x_n y_1 - y_n x_1) \right|$

Where $(x_1, y_1), (x_2, y_2), \dots, (x_n, y_n)$ are the coordinates of the vertices, and $n$ is the number of vertices (in this case, $n = 6$).

Step 3: Perform the Calculation

Now, let’s plug the coordinates into the formula:

Vertices: $(-4, 0), (-2, 3), (2, 3), (4, 0), (2, -3), (-2, -3)$

Shoelace calculation: $\text{Area} = \frac{1}{2} \left| (-4 \times 3) + (-2 \times 3) + (2 \times 0) + (4 \times -3) + (2 \times -3) + (-2 \times 0) \right.$ $\left. - (0 \times -2) - (3 \times 2) - (3 \times 4) - (0 \times 2) - (-3 \times 2) - (-3 \times -4) \right|$

Simplifying: $= \frac{1}{2} \left| (-12 - 6 + 0 - 12 - 6 + 0) - (0 - 6 - 12 + 0 + 6 - 12) \right|$ $= \frac{1}{2} \left| -36 - (-36) \right| = \frac{1}{2} \times 72 = 36$

Thus, the area of the hexagon is 36 square units.

Would you like more details on the calculation or any clarifications?

Here are some follow-up questions you might consider:

1. How does the Shoelace Theorem apply to polygons with different numbers of sides?
2. How can the formula for the area of a regular hexagon differ from this method?
3. What are the advantages of using the Shoelace Theorem for calculating areas?
4. How could this calculation be adapted for 3D shapes?
5. How would the calculation change if the hexagon were irregular?

Tip: The Shoelace Theorem is especially useful when dealing with polygons on a coordinate plane, as it avoids the need for breaking the shape into smaller components.