Shaded Area

As you can see, radius of the bigger circle is $\sqrt{2} +1$

Area of bigger circle= $\pi (\sqrt{2}+1)^2$

Area of 4 smaller circles= $4 \times \pi (1)^2 =4 \pi$

Area of the gap at the centre surrounded by the smaller circles = (Area of square ABCD) - (Area of the four quarters of smaller circles centred at A, B, C and D)

= $2^2- 4 \times \frac{1}{4} \pi (1)^2=4 -\pi$

So, the area of shaded region= (Area of bigger circle) - (Area of 4 smaller circles) - (Area of the gap) = $\pi (\sqrt{2}+1)^2 -4 \pi -(4 -\pi) =4.89$ (approx.)

The centers of the four inner circles forms a square of side length $2$ . The rest of the area are four three-fourths of circles of radius $1$ , so in total that's an area of $3\pi + 4$ .

The radius of the larger circle is one half of the diagonal of the square ( $\sqrt{2}$ ) plus $1$ . Squaring that, multiplying by $\pi$ , and subtracting $3\pi + 4$ yields $4.8857\dots$ as the answer.

Draw the square with its four vertices at the four centers of the $4$ small circles. That square has side equal to $2$ , so its area is $4$ . If you draw the diagonal, the measure is $2\sqrt {2}$ . The segment from the center of the big circle to the center of any of the small circles is $\sqrt {2}$ . Hence, the radii of the big circle is $1 + \sqrt{2}$ , and the area of the entire image is $(3 + 2\sqrt{2})\pi$ . The area of the four small circles is $4\pi$ , and the area of the center of the image is $4 - \pi$ , because on the square we made there were $4$ quarters of the same circle, which is obviously a small circle. Now we subtract...

$(3 + 2\sqrt{2})\pi - 4\pi - (4 - \pi) = 2\sqrt{2}\pi - 4 \approx \boxed {4.89}$ .

my exact calculation went up to.... 4.87324547923062......... :O B-p -.-

Step 1:

First * add the centers * of the little circles. It will give a * square and the length of its side is 2 * adding the radius of two circles each having radius of 1.Now we will find the diagonal of the square and that is * 2 sqrt{2}.**

Now,if we see attentively we clearly see that the diagonal of the square+radius of a small circle +radius of a small circle=Diameter of the big circle . Mathmatically it is * 2 * sqrt{2} +1+1 *.If we divide it by 2 we will get the radius of the big circle.

So,the * radius of the big circle is 1+sqrt{2} . So,the area of the big circle is 3.1416 * [1+sqrt{2}]^2=18.31058666. *

Step 2:

Now the intersection area of the square and each small circle is one 3rd of the area of the small circle.So the remaining area of each small circle is 3/4 of its area.

Each small circle has a area of 3.1416 * 1^2=3.1416 and 3/4 of this is 3.1416 * [3/4].

There are four small circles. * So we have to multiply this by 4 which results in 3 * 3.1416. *

The area of the square is 4.

So, the area covered by the square and small circles is 4+3 * 3.1416.

Now we have to substract this from the area of the big circle to get the area of the shaded area.

So, 18.31058666-3 * 3.1416-4=4.89

This is our desired area and the answer is 4.89

At first draw straight lines from centre of each small circles to make a square of side 2 units(within the bigger circle). Calculate the diagonal of the square formed using formula (2)^1/2 * a (sorry i don't have the root sign ) then extend the diagonal so that it touches the bigger circle . So we get radius of the bigger circle which is 2.414 units. Now the area of the square is 4 sq.units and the net area of small circles is 3 pi 1 sq units. Now add this area to square's area and subtract it from the area of the bigger circle's area i.e.(18.3105 - 13.4247)=4.8858 = 4.89 .

4.855 sq units

1. Construct a square by joining the centers of 4 small circles. Diagonal of square = 2 sqrt(2) and Diameter of large circle = Diagonal of square + 2 radius of small circle = 4.828
2. Using step 1, Calculate area of large circle and subtract area of square and area of 3/4s of 4 small circles to get the area of shaded portion.

alt text

Let the radius of the smaller circles be $\displaystyle r$ .
Let the radius of the larger circle be $\displaystyle R$

From the figure, clearly,

$2R = 2r + \sqrt{(2r)^2+(2r)^2} = 2r(\sqrt{2}+1)$

$R = (\sqrt{2}+1)r$

Now,
Area of the unshaded region $=$ Area of square formed by joining the centers of the circles $+ 4(\frac{3}{4}) \pi r^2$
Area of the unshaded region $= 4r^2 + 3\pi r^2$

Hence,
Area of the shaded region $= \pi R^2 -$ Area of the unshaded region

Area of the shaded region $= \boxed{4.89sq.units}$