Circles Abound

From the figure above, it is clear that the area of the green region is the area of the $2 \times 2$ square minus the area of 4 quadrants of radius 1 or a circle of radius 1, or $A = 2 \times 2 - \pi \times 1^2 = \boxed{4-\pi}$ .

We get a square of side 2 units by joining all the centres of the circle.

The square has yellow part & green part.

The yellow part consists of 4 congruent quadrants of a circle. (They are congruent because they all have central angle 90 and radius 1)

Hence, the area of yellow part = Area of circle with radius 1 = π * 1 * 1 = π

Area of the square = side * side = 4

Therefore, Green region = Area of square - Area of yellow region = 4 - π

If we join the centres of all circles , we will get a square with side = 2×radius of circle = 2 | Area of square = Area of four quadrants + Green Area | Since all circles are congruent , the four quadrants will make up a complete circle with radius = 1 | 2×2 = π×1×1 + Green Area | Therefore , Green Area = 4-π . ||

From this image we will only focus on what is inside the 2x2 square. We can see that the areas of the 4 quarter circles are each 1/4 π, so the total of those areas is π. The area of the square is 4, so the area of the green region is 4-π .

Focusing on one quadrant: $\text{Green Area}=\frac{\text{Square of sidelength } 2-\text{circle of radius }1}{4}$

If we consider the entire diagram, the total green area is 4 times this amount, and is therefore a $\text{Square of sidelength }2-\text{circle of radius }1=2^2-\pi=\boxed{4-\pi}$ .

Relevant wiki: Length and Area Problem Solving

Clearly from the diagram,

Area of Green Region=Area of Red Square - Area of the Circle Inside the Red Square= $2^2-(π \times 1 \times 1 )= 4 - π$

As each circle has radius 1, area of all the 4 circles= $\pi r^2*4$ = $4\pi$ . If we make a square on the boundary of the figure, we can notice that $1$ more the same shape as the green region can be contained in the square. Thus, area of green region= $\frac{16-4\pi}{2}$ , which can be reduced to $4-\pi$ . That is the answer!

Area of green region = area of square - area of circle = $2^2 - π(1^2) = 4 - π$

Make a square of 4 by 4 units around the four circles, let the required green area be =x
So now, area of square - area of circle=4x
As we can see from symmetricity that the part between any two circle and the edge of the square has an area x/2 and there are 4 such parts so these parts have area 2x and the part between a circle and the corner of the square has an area x/4 and there are 4 such parts so they give an area x,
Now, area of square - area of circle = 16-4π
=x(green part) + 2x +x = 4x
So, x= (16-4π)/4= 4-π
Which is the required area.

Area=(2)^2-4*(pi\4)

The green area is equivalent to the bits of area between the unit circle and the square subscribing it.

Easy.

Area of each circle = 1*∏ = ∏

Circumscribe one of the circles with a square with side = 2

Area of the square = 2*2=4

Area of that part of the square outside the circle = 4-∏

Area of the green area inside the square = (4-∏)/4

True for all four circles

Therefore the combined green area = 4*(4-∏)/4 = 4-∏

There is four circle having same radius so we can have area of square by joining there circle is=4,and every circle contribute his 1/4th part so area of one circle is=3.14/4 (2)^2,and area covered by four circle in square is 4 (area of one circle covered) is directly=3.14 so we can write the soution as=4-3.14.hence solved.