Try this beautiful problem from Geometry based Largest area.
Largest area – AMC-8, 2003 – Problem 22
The following figures are composed of squares and circles. Which figure has a shaded region with largest area?

- $A$
- $B$
- $C$
Key Concepts
Geometry
Circle
Square
Check the Answer
But try the problem first…
Answer:$C$
AMC-8 (2003) Problem 22
Pre College Mathematics
Try with Hints
First hint
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 ……….
Second Hint
area of circle =\(\pi r^2\)
can you finish the problem……..
Final Step

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
Other useful links
- https://www.cheenta.com/triangle-and-quadrilateral-amc-10a-2005-problem-25-2/
- https://www.youtube.com/watch?v=oUyHFKVB9IY