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To calculate the area of the polygon in the given diagram, we need to divide the shape into simpler geometric figures. Here's a step-by-step breakdown of the approach:

### Step 1: Identify key geometric components
The shape consists of:
- Two semicircles at the top-left and top-right corners.
- A large central rectangle in the middle.
- Two triangular notches at the bottom.

Each small square on the grid is \(4 \, \text{cm} \times 4 \, \text{cm}\), so we can now compute the dimensions of each part.

### Step 2: Measure dimensions from the grid
By counting the grid squares:
- **Radius of each semicircle**: 2 squares or \( 8 \, \text{cm} \) (since each square is 4 cm).
- **Width of the central rectangle**: 8 squares or \( 32 \, \text{cm} \).
- **Height of the rectangle**: 3 squares or \( 12 \, \text{cm} \).
- **Width of the triangular notches**: 2 squares or \( 8 \, \text{cm} \) each.
- **Height of the triangular notches**: 2 squares or \( 8 \, \text{cm} \).

### Step 3: Calculate areas
Now, we compute the areas of each component:

1. **Area of each semicircle**:
   \[
   A_{\text{semicircle}} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (8)^2 = 32 \pi \, \text{cm}^2
   \]
   Total area for two semicircles:
   \[
   A_{\text{semicircles}} = 2 \times 32 \pi \approx 201.06 \, \text{cm}^2
   \]

2. **Area of the central rectangle**:
   \[
   A_{\text{rectangle}} = \text{width} \times \text{height} = 32 \times 12 = 384 \, \text{cm}^2
   \]

3. **Area of the two triangular notches**:
   The area of one triangle:
   \[
   A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 8 \times 8 = 32 \, \text{cm}^2
   \]
   Total area for two triangles:
   \[
   A_{\text{triangles}} = 2 \times 32 = 64 \, \text{cm}^2
   \]

### Step 4: Calculate the total area
Now, we sum the areas and subtract the area of the triangles (since they are cutouts):
\[
A_{\text{total}} = A_{\text{semicircles}} + A_{\text{rectangle}} - A_{\text{triangles}}
\]
\[
A_{\text{total}} = 201.06 + 384 - 64 = 521.06 \, \text{cm}^2
\]

Thus, the total area of the polygon is approximately **521.06 cm²**.

Would you like a more detailed breakdown of any specific part, or have further questions?

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Here are 5 related questions:
1. How would the area change if the squares were smaller, say 2 cm × 2 cm?
2. Can you calculate the perimeter of the shape using the same grid?
3. How would the total area differ if the semicircles were complete circles?
4. What is the ratio of the area of the triangular notches to the total area?
5. How could this problem be solved using integration for more complex shapes?

**Tip**: When dividing irregular shapes into smaller parts, always check if symmetry can help simplify the calculations!

Berechnen Sie den Flächeninhalt des Vielecks, wenn ein Kästchen 4 cm groß ist. Runden Sie das Ergebnis auf zwei Nachkommastellen.

Berechnen Sie den Flächeninhalt eines Vielecks mit zwei Halbkugeln, einem Rechteck und Dreiecksausschnitten, basierend auf einem Gitter mit Kästchen, die 4 cm groß sind. Runden Sie das Ergebnis auf zwei Nachkommastellen.

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