Find the area of the shaded region in the figure, where ABCD is a square of side 10 cm, and semicircles are drawn with each side of the square as diameter (use π = 3.14).

Steps to Solve

1. Calculate the area of the square

The area $A$ of the square with side length $s = 10 , \text{cm}$ is calculated as: $$ A = s^2 = 10^2 = 100 , \text{cm}^2 $$

2. Find the radius of the semicircles

Each semicircle has a diameter equal to the side of the square, which is $10 , \text{cm}$. Thus, the radius $r$ is: $$ r = \frac{10}{2} = 5 , \text{cm} $$

3. Calculate the area of one semicircle

The area $A_{semi}$ of one semicircle is given by: $$ A_{semi} = \frac{1}{2} \pi r^2 = \frac{1}{2} \times 3.14 \times 5^2 = \frac{1}{2} \times 3.14 \times 25 = 39.25 , \text{cm}^2 $$

4. Calculate the total area of the four semicircles

Since there are four semicircles, the total area $A_{total_semi}$ is: $$ A_{total_semi} = 4 \times A_{semi} = 4 \times 39.25 = 157 , \text{cm}^2 $$

5. Find the area of the shaded region

The area of the shaded region is the area of the square minus the total area of the semicircles: $$ A_{shaded} = A_{square} - A_{total_semi} = 100 - 157 = -57 , \text{cm}^2 $$

Since this result does not make sense, it indicates that the semicircles double back into the square and create an overlap instead.