The image presents a problem where a circle is inscribed in a square, and the goal is to calculate the total area of the four shaded corner sections outside the circle but within the square. The radius of the circle is given as $62 \, \text{cm}$.

Let's break it down step by step:

Step 1: Calculate the side length of the square

Since the circle touches the midpoint of each side of the square, the diameter of the circle is equal to the side length of the square.

• The diameter of the circle is twice the radius: $\text{Diameter of circle} = 2 \times 62 = 124 \, \text{cm}$ Thus, the side length of the square is also $124 \, \text{cm}$.

Step 2: Calculate the area of the square

The area $A_s$ of a square is given by the formula: $A_s = \text{side length}^2$ So, the area of the square is: $A_s = 124^2 = 15376 \, \text{cm}^2$

Step 3: Calculate the area of the circle

The area $A_c$ of a circle is given by the formula: $A_c = \pi \times r^2$ where $r = 62 \, \text{cm}$.

$A_c = \pi \times 62^2 = \pi \times 3844 \approx 12084.78 \, \text{cm}^2$

Step 4: Calculate the area of the shaded regions

The shaded regions are the parts of the square not occupied by the circle. The total area of the shaded regions is the difference between the area of the square and the area of the circle.

$\text{Area of shaded regions} = A_s - A_c$ $\text{Area of shaded regions} = 15376 - 12084.78 \approx 3291.22 \, \text{cm}^2$

Final Answer:

The total area of the four shaded sections is approximately $3291.2 \, \text{cm}^2$ (to 1 decimal place).

Would you like more details on any part of this solution? Here are some follow-up questions that might interest you:

1. How is the relationship between the square's side length and the circle's diameter derived?
2. Can this method be applied to other shapes (e.g., a triangle inscribed in a circle)?
3. What if the radius of the circle was different? How would that affect the shaded area?
4. How does changing the precision of $\pi$ affect the result?
5. What happens to the shaded area if the circle does not perfectly touch the square's midpoint?

Tip: Always make sure to double-check units and rounding, especially when dealing with area or volume problems.