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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

But try the problem first…

Answer:$C$

Source

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=$\pi (1)^2-(\sqrt 2)^2$=$\pi – 2$