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The problem presented involves calculating the gray-shaded area in a geometric figure that includes a circle inscribed inside an irregular pentagon. Here's a breakdown of how to solve this problem:

### Data:
- **Diameter of the circle**: 3 cm (which gives us a radius of 1.5 cm).
- **Area of the irregular pentagon** \( GHEFD \): 8.83 cm².
- The formulas for circumference and area are given:
  - \( \pi = 3.14 \)
  - Perimeter of the circle \( P = 2\pi r \)
  - Area of the circle \( A = \pi r^2 \)

### To solve the problem:
1. **Calculate the area of the circle**:
   - Using the formula \( A = \pi r^2 \), where \( r = 1.5 \) cm.
   - This will give you the area of the circle.

2. **Subtract the area of the circle from the area of the pentagon**:
   - The gray-shaded area is the part of the pentagon that remains after removing the area of the inscribed circle.
   - So, subtract the area of the circle from the given area of the pentagon.

3. **Result**: The result of this subtraction will give you the gray-shaded area.

Would you like a detailed step-by-step calculation or further clarifications?

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Here are five related questions to expand on this:
1. How would the problem change if the diameter of the circle were larger?
2. What is the formula for the circumference of a circle, and how would it help in other geometry problems?
3. How do you find the area of other irregular polygons?
4. Can you inscribe a circle in other polygons, and what conditions are necessary for that?
5. How would you calculate the area if the pentagon were a regular pentagon?

**Tip**: Always double-check whether the given area (such as for the pentagon) includes or excludes inner shapes like inscribed circles to avoid mistakes in your calculations.

¿Cuánto mide el área de la parte del pentágono irregular GFHECD que queda de color gris en la figura?

Learn how to calculate the gray-shaded area in an irregular pentagon with an inscribed circle. This problem involves geometry concepts such as area calculation and polygon properties, suitable for Grades 6-8.

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