In a circular park of diameter 80m, there is a square-shaped playground of maximum area. The area of the playground is

Options

A. `3200m^(2)`
B. `6400m^(2)`
C. `1600m^(2)`
D. `12800m^(2)`

Answer

To find the area of the square-shaped playground that fits inside a circular park with a diameter of 80 meters, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Diameter of the Circle**:
The diameter of the circular park is given as 80 meters.

2. **Determine the Radius of the Circle**:
The radius (r) is half of the diameter.
\[ r = \frac{80}{2} = 40 \text{ meters} \]

3. **Relate the Square's Diagonal to the Circle's Diameter**:
The diagonal of the square will be equal to the diameter of the circle. Therefore, the diagonal (d) of the square is 80 meters.

4. **Use the Relationship Between the Side of the Square and Its Diagonal**:
For a square with side length \( a \), the relationship between the side and the diagonal is given by:
\[ d = a\sqrt{2} \]
Setting the diagonal equal to the diameter of the circle:
\[ a\sqrt{2} = 80 \]

5. **Solve for the Side Length of the Square**:
Rearranging the equation to find \( a \):
\[ a = \frac{80}{\sqrt{2}} = \frac{80\sqrt{2}}{2} = 40\sqrt{2} \text{ meters} \]

6. **Calculate the Area of the Square**:
The area (A) of the square is given by:
\[ A = a^2 \]
Substituting the value of \( a \):
\[ A = (40\sqrt{2})^2 = 40^2 \cdot 2 = 1600 \cdot 2 = 3200 \text{ square meters} \]

### Final Answer:
The area of the playground is **3200 square meters**.

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