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Try this beautiful problem from Geometry based on Intersection of two Squares.

Two \(4\times 4\) squares intersect at right angles, bisecting their intersecting sides, as shown. The circle's diameter is the segment between the two points of intersection. What is the area of the shaded region created by removing the circle from the squares?

- \(28-2\pi\)
- \(25-2\pi\)
- \(30-2\pi\)

Geometry

square

Circle

But try the problem first...

Answer: \(28-2\pi\)

Source

Suggested Reading

AMC-8, 2004 problem 25

Pre College Mathematics

First hint

Area of the square is \(\pi (r)^2\),where \(r\)=radius of the circle

Can you now finish the problem ..........

Second Hint

Clearly, if 2 squares intersect, it would be a smaller square with half the side length, 2.

can you finish the problem........

Final Step

Clearly the intersection of 2 squares would be a smaller square with half the side length, 2.

The area of this region =Total area of larger two squares - the area of the intersection, the smaller square i.e \(4^2+4^2 -2^2=28 \)

Now The diagonal of this smaller square created by connecting the two points of intersection of the squares is the diameter of the circle

Using the Pythagorean th. diameter of the circle be \(\sqrt{2^2 +2^2}=2\sqrt 2\)

Radius=\(\sqrt 2\)

area of the square=\(\pi (\sqrt2)^2\)=\(2\pi\)

Area of the shaded region= 28-2\(\pi\)

- https://www.cheenta.com/area-of-square-and-circle-amc-8-2011-problem-25/
- https://www.youtube.com/watch?v=BXWGKh01nOw

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