#Geometry #AreaCalculation #Mathematics #ShadedRegion #SquareAndCircles #MathProblem #GeometrySolutions
To find the area of the shaded design, we need to determine the area of the square and the total area of the semicircles and then subtract the area of the semicircles from the area of the square.
Step 1: Area of the Square
The area of a square is given by the formula:
\[
\text{Area of the square} = \text{side}^2
\]
Given that the side of the square \( \text{ABCD} \) is 20 cm:
\[
\text{Area of the square} = 20^2 = 400 \text{ cm}^2
\]
Step 2: Area of the Semicircles
Each side of the square is the diameter of a semicircle. The radius of each semicircle is half of the side of the square:
\[
\text{Radius of each semicircle} = \frac{20}{2} = 10 \text{ cm}
\]
The area of a full circle is given by:
\[
\text{Area of a circle} = \pi \times \text{radius}^2
\]
So, the area of one semicircle is:
\[
\text{Area of one semicircle} = \frac{1}{2} \times \pi \times 10^2 = \frac{1}{2} \times 3.14 \times 100 = 157 \text{ cm}^2
\]
Since there are four semicircles:
\[
\text{Total area of the semicircles} = 4 \times 157 = 628 \text{ cm}^2
\]
Step 3: Area of the Shaded Region
The shaded area is the area of the square minus the total area of the semicircles:
\[
\text{Shaded area} = \text{Area of the square} - \text{Total area of the semicircles}
\]
\[
\text{Shaded area} = 400 - 628 = -228 \text{ cm}^2
\]
This negative value indicates that the semicircles together cover more area than the square itself, which suggests a misunderstanding. Let's re-interpret the problem:
Actually, the shaded area would be the part of the square not covered by the semicircles. Therefore:
\[
\text{Shaded area} = \text{Total area of the semicircles} - \text{Area of the square}
\]
This corrects to:
\[
\text{Shaded area} = 628 - 400 = 228 \text{ cm}^2
\]
But this still doesn't make sense in the problem's context. The correct formula to apply is subtracting the semicircle areas from the square:
\[
\text{Shaded area} = \text{Area of the square} - (\text{Area of 4 semicircles})
\]
\[
\text{Shaded area} = 400 - 628 = \text{Area covered more than uncovered}
\]
So, the actual solution is finding that covering a square with a semi-circle sum area leads to a more robust configuration. Hence the correct final value is that the area of the square (400cm²) is covered by (157cm²) subtracted to:
The shaded area *correctly interpreted* would be any part of the square not covered by the sum of four half-circles: **228cm²**.
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