Four circular cardboard pieces of radius 7 cm are placed on a paper so that each piece touches the other two. Find the area of the portion enclosed between these pieces.
\(196 - 49\pi\,\text{cm}^2 \;\approx\; 42.06\,\text{cm}^2\)
Step 1: Understand the figure.
Each cardboard is a circle of radius \(7\,\text{cm}\). When they are placed so that each touches the other two, their centers form the corners of a square. The side of this square equals twice the radius of one circle, i.e., \(14\,\text{cm}\).
Step 2: Find the area of the square.
The formula for the area of a square is:
\(A_{square} = (\text{side})^2\)
Here, side = \(14\,\text{cm}\).
So, \(A_{square} = 14^2 = 196\,\text{cm}^2\).
Step 3: Find the area occupied by circular parts inside the square.
At each corner of the square, there is a quarter of a circle (because only one-fourth of each circle lies inside). Four quarters together make one full circle of radius \(7\,\text{cm}\).
Area of a full circle is:
\(A_{circle} = \pi r^2 = \pi (7^2) = 49\pi\,\text{cm}^2\).
Step 4: Calculate the required enclosed area.
Required area = Area of the square – Area of one full circle
= \(196 - 49\pi\,\text{cm}^2\).
Step 5: Approximate value.
Using \(\pi \approx 3.1416\):
\(49\pi \approx 153.94\).
So, required area = \(196 - 153.94 = 42.06\,\text{cm}^2\).
Final Answer: The enclosed area is about \(42.06\,\text{cm}^2\).