The image shows a rectangle $ABCD$ with dimensions $36 \, \text{cm}$ by $28 \, \text{cm}$. Inside the rectangle, there are 4 congruent semicircles whose flat sides touch the midpoints of the sides of the rectangle. The semicircles are connected at the center, forming a shape. The goal is to find the area of the shaded region, which is the area of the rectangle minus the area of the combined semicircles.

Step 1: Area of the rectangle

The area of the rectangle is simply: $\text{Area of rectangle} = \text{Length} \times \text{Width} = 36 \, \text{cm} \times 28 \, \text{cm} = 1008 \, \text{cm}^2$

Step 2: Radius of each semicircle

From the image, it’s given that the total width of the rectangle is 36 cm, and two semicircles span horizontally across the width of the rectangle. Each semicircle has a radius of $10 \, \text{cm}$ (since two times the radius gives the total width of 20 cm for the semicircles).

Thus, the radius $r$ of each semicircle is $10 \, \text{cm}$.

Step 3: Area of the combined semicircles

Each full circle can be constructed by combining two semicircles. Since we have 4 semicircles, these form 2 full circles. The area of one full circle is given by: $\text{Area of one circle} = \pi r^2 = \pi (10)^2 = 100\pi \, \text{cm}^2$ Thus, the total area of the two full circles is: $\text{Total area of circles} = 2 \times 100\pi = 200\pi \, \text{cm}^2$ Approximating $\pi \approx 3.1416$, we get: $\text{Total area of circles} \approx 200 \times 3.1416 = 628.32 \, \text{cm}^2$

Step 4: Area of the shaded region

The area of the shaded region is the area of the rectangle minus the area of the circles: $\text{Area of shaded region} = \text{Area of rectangle} - \text{Area of circles} = 1008 \, \text{cm}^2 - 628.32 \, \text{cm}^2 = 379.68 \, \text{cm}^2$

Thus, the area of the shaded region is approximately $\boxed{379.68 \, \text{cm}^2}$.

Would you like any further clarifications or details on this solution?

Related questions:

1. How would the solution change if the semicircles had different radii?
2. What would be the perimeter of the shaded region?
3. How would you calculate the area if the shape in the center was an ellipse instead of circles?
4. What is the total perimeter of the combined semicircles?
5. How would the area calculation change if the semicircles were inscribed inside a square?

Tip: Always double-check the given dimensions in a geometric problem to ensure that your radius and calculations align with the problem's configuration.