Try this beautiful problem from Geometry based Largest area.
The following figures are composed of squares and circles. Which figure has a shaded region with largest area?
Geometry
Circle
Square
Answer:$C$
AMC-8 (2003) Problem 22
Pre College Mathematics
To find out the largest area at first we have to find out the radius of the circles . all the circles are inscribed ito the squares .now there is a relation between the radius and the side length of the squares....
Can you now finish the problem ..........
area of circle =\(\pi r^2\)
can you finish the problem........
In A:
Total area of the square =\(2^2=4\)
Now the radius of the inscribed be 1(as the diameter of circle = side length of the side =2)
Area of the inscribed circle is \(\pi (1)^2=\pi\)
Therefore the shaded area =\(4- \pi\)
In B:
Total area of the square =\(2^2=4\)
There are 4 circle and radius of one circle be \(\frac{1}{2}\)
Total area pf 4 circles be \(4 \times \pi \times (\frac{1}{2})^2=\pi\)
Therefore the shaded area =\(4- \pi\)
In C:
Total area of the square =\(2^2=4\)
Now the length of the diameter = length of the diagonal of the square=2
Therefore radius of the circle=\(\pi\) and lengthe of the side of the square=\(\sqrt 2\)
Thertefore area of the shaded region=Area of the square-Area of the circle=\(\pi (1)^2-(\sqrt 2)^2\)=\(\pi – 2\)
Therefore the answer is C
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