Let's break down the problem and solve it step by step.

Given:

• The radius of the circle is $r = 82 \, \text{cm}$.
• The circle is inscribed in a square, and the edges of the circle touch the midpoints of each side of the square.

Objective:

We need to calculate the total area of the four shaded sections, which are the regions outside the circle but inside the square.

Steps:

1. Calculate the area of the circle: The area of a circle is given by the formula: $A_{\text{circle}} = \pi r^2$ Substituting $r = 82 \, \text{cm}$, we can calculate this.

2. Calculate the side length of the square: Since the circle touches the midpoint of each side, the diameter of the circle is equal to the side length of the square. The diameter is: $\text{Diameter} = 2r = 2 \times 82 = 164 \, \text{cm}$ So, the side length of the square is 164 cm.

3. Calculate the area of the square: The area of the square is: $A_{\text{square}} = \text{side}^2 = 164^2$

4. Calculate the area of the shaded regions: The total shaded area is the difference between the area of the square and the area of the circle: $A_{\text{shaded}} = A_{\text{square}} - A_{\text{circle}}$

Now, let's calculate these values.The total area of the four shaded sections is $5771.9 \, \text{cm}^2$, rounded to 1 decimal place.

Would you like further details or have any questions about this solution?

Here are some related questions for further exploration:

1. What would the area of the shaded regions be if the radius of the circle were doubled?
2. How would the area change if the square were replaced by a rectangle of different dimensions?
3. Can we calculate the perimeter of the shaded regions directly?
4. How would the problem change if the circle were not perfectly inscribed but offset inside the square?
5. How is the area ratio between the circle and square affected by increasing the circle's radius?

Tip: Always check if the radius given is correct, as it directly affects both the circle's area and the square's side length.