https://r2.erweima.ai/mathbot/image/1726026876224o4jsupsd.jpg

Let's solve each part of the problem step-by-step.

### a) Calculation of the Gray Area

In part **a**, we have a square with a quarter circle inside it. The radius of the quarter circle is \( r = 5 \).

1. **Area of the square:**
   The side length of the square is equal to the diameter of the quarter circle. Since the radius \( r = 5 \), the side length is:
   \[
   \text{Side length} = 2 \times r = 2 \times 5 = 10
   \]
   The area of the square is:
   \[
   \text{Area of the square} = \text{side}^2 = 10^2 = 100
   \]

2. **Area of the quarter circle:**
   The area of a full circle is given by:
   \[
   \text{Area of the circle} = \pi r^2 = \pi \times 5^2 = 25\pi
   \]
   Since we only have a quarter of the circle:
   \[
   \text{Area of the quarter circle} = \frac{1}{4} \times 25\pi = 6.25\pi
   \]

3. **Gray area calculation:**
   The gray area is the area of the square minus the area of the quarter circle:
   \[
   \text{Gray area} = 100 - 6.25\pi
   \]
   Approximating \(\pi \approx 3.14\):
   \[
   \text{Gray area} \approx 100 - 6.25 \times 3.14 \approx 100 - 19.625 = 80.375
   \]

### b) Calculation of the Gray Area

In part **b**, we have a semicircle with a triangle inscribed in it.

1. **Area of the semicircle:**
   The diameter of the semicircle is \( d = 12 \), so the radius is:
   \[
   r = \frac{d}{2} = \frac{12}{2} = 6
   \]
   The area of a full circle would be:
   \[
   \text{Area of the full circle} = \pi r^2 = \pi \times 6^2 = 36\pi
   \]
   The area of the semicircle is half of the full circle:
   \[
   \text{Area of the semicircle} = \frac{1}{2} \times 36\pi = 18\pi
   \]

2. **Area of the triangle:**
   The base of the triangle is equal to the diameter of the semicircle, \( d = 12 \). The height of the triangle is equal to the radius of the semicircle, \( r = 6 \).
   The area of the triangle is:
   \[
   \text{Area of the triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 12 \times 6 = 36
   \]

3. **Gray area calculation:**
   The gray area is the area of the semicircle minus the area of the triangle:
   \[
   \text{Gray area} = 18\pi - 36
   \]
   Approximating \(\pi \approx 3.14\):
   \[
   \text{Gray area} \approx 18 \times 3.14 - 36 = 56.52 - 36 = 20.52
   \]

### c) Calculation of the Gray Area

In part **c**, we have an equilateral triangle with a circle inscribed in it.

1. **Area of the equilateral triangle:**
   The side length of the triangle is \( s = 6 \). The formula for the area of an equilateral triangle is:
   \[
   \text{Area of the triangle} = \frac{\sqrt{3}}{4} s^2 = \frac{\sqrt{3}}{4} \times 6^2 = \frac{\sqrt{3}}{4} \times 36 = 9\sqrt{3}
   \]
   Approximating \(\sqrt{3} \approx 1.732\):
   \[
   \text{Area of the triangle} \approx 9 \times 1.732 = 15.588
   \]

2. **Area of the inscribed circle:**
   The radius \( r \) of a circle inscribed in an equilateral triangle is given by:
   \[
   r = \frac{s \sqrt{3}}{6}
   \]
   Substituting \( s = 6 \):
   \[
   r = \frac{6 \sqrt{3}}{6} = \sqrt{3}
   \]
   The area of the circle is:
   \[
   \text{Area of the circle} = \pi r^2 = \pi (\sqrt{3})^2 = 3\pi
   \]
   Approximating \(\pi \approx 3.14\):
   \[
   \text{Area of the circle} \approx 3 \times 3.14 = 9.42
   \]

3. **Gray area calculation:**
   The gray area is the area of the triangle minus the area of the inscribed circle:
   \[
   \text{Gray area} = 9\sqrt{3} - 3\pi
   \]
   Substituting the approximations:
   \[
   \text{Gray area} \approx 15.588 - 9.42 = 6.168
   \]

### Summary of Results

- **a)** Gray area ≈ 80.375
- **b)** Gray area ≈ 20.52
- **c)** Gray area ≈ 6.168

Would you like further details on any of these calculations or have any questions? 

### Related Questions

1. How do you find the area of a sector of a circle?
2. How do you calculate the area of an inscribed polygon?
3. What is the formula for the area of an equilateral triangle?
4. How do you approximate the value of \(\pi\) for calculations?
5. Can you derive the formula for the radius of an inscribed circle in an equilateral triangle?

### Tip

Remember, understanding the relationship between different geometric shapes helps simplify complex area calculations!

Learn how to calculate the gray areas of composite shapes including squares with quarter circles, semicircles with inscribed triangles, and equilateral triangles with inscribed circles. This problem involves key geometry concepts and formulas suitable for high school students.

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