Here's how I solved it:

The length of the diagonal of a square is equal to $x\sqrt{2}$ , with x being equivalent to the side lengths. Now, if you were to rotate the square $45 ^ \circ$ about the center of the circle (and its own center), then the length of the diagonal would become the diameter of the circle. This means that the diameter of the circle the original square is inscribed in is equal to $5\sqrt{2}$ , with 5 obviously being the side lengths since it is equal to $\sqrt{25}$ , the area. Looking at the diagram, the diameter of the circle is quite obviously the side length of the square outside it. So now we are on the second square, and we know one side is $5\sqrt{2}$ units long. We can repeat the first step and say the diagonal of this square is equal to $5\sqrt{2}*\sqrt{2}$ , which is, of course, $5*2$ , or $10$ . Now, following the first step, we can say that the diameter of the circle outside this square is also equal to $10$ units. The diameter of this circle is, once again, equal to one side length of Square ABCD. So if $s=d$ , we can use the substitution property to say that $s=10$ . We are finding the area, so we just have to find $10^2$ , and that is the answer!

You guys are giving way to complex explanations . Here's a really simple one: The shaded square is inside the bigger one 4 times. So to find out the area of the bigger one all you do is 4*25=100

Each side of the successive square is equal to the diagonal of the previous square This gives side of ABCD as 10 and area as 100

a" a =25 =>a"=5. Diagonal of this smallest square is=(5^2+5^2)^(1/2). and (5^2+5^2)^(1/2) it diameter of the smallest circle. so it is the side of the 2nd squre. thus we find AB=10. so ABCD=10*10=100.

I used the Pythagorean Theorem to find the diagonal of the first square, which will be the side of the next square. I repeated this with the other two squares. The sides of the shaded square were 5, the sides of the second square were 5squareroot2, and the sides of the outer square were 10. 10*10=100, which is the answer.

$25=\frac{d^2}{2}$ $d^2=50\;so\;d=\sqrt{50}=5\sqrt{2}$ The diagonal of the red square is the diameter of the 1st circle so it is the side length of the next square from there we get the diagonal as $\sqrt{2(5\sqrt{2})^2}=10$ which is again the diameter of the biggest circle and which is also the side of the square ABCD .So area of square ABCD= $l^2=10^2=\boxed{100\;units}$

Guess a multiple of 25.

YAYme!

now, im gonna learn how to really do it.

√25=5 arms of shaded area, so √(5^2 5^2)= 7.07. Which is Dia of circula . Dia of circula = length of square= 7.07. so, √(7.07^2 7.07^2)= 10. Which is Dia of circula . Dia of circula = length of square=10. so area of square is 10*10=100.

first of all, calculate the one side of shaded squarewhich is nothing but 5units. then by Pythagoras theorem, calculate the diameter of inner circle. we will get 5X(2)^(1/2). if you see the figure carefully, the diameter of the inner circle is nothing but the side of 2nd square. once again calculate the diameter of second circle by taking the sides of 2nd square using Pythagoras theorem, we will get 10units. now this diameter of 2nd circle gives the sides of third and the required side to calculate the area of square ABCD. apply the square formula as 10X10=100

the square formed by joining the midpoints of another bigger square has an area half that of bigger square the first inner square has an area half of ABCD and the shaded square is half that of the first inner square which implies the area of the shaded region is one fourth part of ABCD area

Since we could turn $45^\circ$ ,we recognize that the red square is half of the inner square and the inner square is half of the ABCD square so the answer is $25 \times \frac{1}{(\frac{1}{2})^2}=\boxed{\large{100}}$

The diagonal of a given square becomes the side of the next square. Hence the side of the second square is 5sqrt(2), with diagonal (5(sqrt(2))sqrt(2), so the third square has a side of 10, area = 100. Ed Gray

turn the red square 45 degrees it fits the next square across perfectly (because the circle fits) this makes the second square become the red diamond and four white triangles the four triangles if folded over will fit the red square exactly so the second square is twice as big as the first one 2x25=50 repeat for the next square up 2x50=100

the diagonal of the small square = the side of the middle square.=7.07. to find the diagonal of the middle square we multiply 7.07 by the square root of 2 which equals 9.99. We now have the side of square ABCD. 9.99^2= 99.96 when rounded equals 100.

Let sides of outer, inner and inner(est) squares be $a, b, c$ respectively. Diagonal of $c$ 's square is = diameter of $c$ 's outer circle = side of $b$ . Therefore using pythagorean theorem,

$b=\sqrt [ 2 ]{ 50 }$

Diagonal of $b$ is = diameter of $b$ 's outer circle = side of $a$

$a=\sqrt [ 2 ]{ 100 } \\ a=10\\ \\ \therefore { a }^{ 2 }=100$

Let $a_1 = 25 =$ Area of shaded square and $a_2 =$ Area of $\overline{ABCD}$

Since the shape is a square it's area is equal to it's length squared so the length is

$\sqrt{25} = 5$

The corners of the square are touching the edges of the first circle ( $c_1$ ) so the diameter of the circle can be found using Pythagorean theorem (this is because the diagonal makes a right angled triangle with the length)

$c_1 = \sqrt{5^2 + 5^2} = 5\sqrt{2}$

This is also conveniently the length of the second square which can be used to find the diameter of the second circle ( $c_2$ )

$c_2 = \sqrt{2(5\sqrt{2})^2} = 10$

The same rules apply for the square $\overline{ABCD}$ so $10$ is the length which makes the area

$10^2 = 100 = \overline{ABCD}$

The area of the middle square should be the diagonal of the red square, squared. You can visualize this by spinning the red square 45 degrees so that the vertices will touch the center of either side on the larger square. So the diagonal is sqrt(5^2+5^2)=sqrt(50). Squaring this gets 50. You may notice that the area of the larger square is 2x the area of the smaller square. Using the same concept, the area of the largest circle is 50*2=100.

it was pretty easy, but here is how i solved it: all i did was 25 x 4= 100, and the bigger square is at least 4 times that size.

Since the square has gone through 2 circles the ratio of the shaded square to ABCD is 1:4

it is a easy problem even a kid can solve it.........

Area of square=5x5=25 side of square=diameter of circle diagonal of 1 st square(shaded)is the diameter of 1 st circle 1)1 st diagonal =√(25+25)=√50 diagonal of 2nd square is the diameter of 2 st circle 2)2 nd diagonal=√(50+50)=√100=10 therefore area ABCD=10x10=100 unit square

i just guessed it. works all the time

5 square is 25 , after that radius is5root2 after that 10 then area is 100

Guys... I see people writing such elaborate guides and I'm just over here estimating like what?!

Anyway, I got 100. I just took the red square, and guessed it fit inside the big square like four times. So I thought 100 because 25 times 4 is 100.

Use a bunch of \sqrt2's and then u can solve it

My solution for this problem is: The diagonal length of the shaded square is equal to the diameter of the circle it is in, therefore the diagonal length is equal to the length, thus height of the square outside it. By imagining the square tilted 90°, we can tell that the first outer square from the shaded on is double the area. Repeat the process by replacing the shaded square and the middle square with the middle and outermost squares respectively.

By default, we would get Area of shaded square, (25) 2 2 = 25*4 = 100 units^2

An alternative would be this:

When the shaded square is rotated 90°, the diagonal length becomes equal to the diameter of the circle around it. So, with the shaded square's area being 25, the length of each side is 5. Using Pythagoras' theorem where a^2+b^2=c^2,

5^2+5^2=25+25=50=c^2

C= sqrt50

When the length of the middle square is sqrt50, repeat the calculating process by using Pythagoras' theorem

sqrt50^2+sqrt50^2=50+50=100=c^2

C is proven to be 10 when c=sqrt100=10. As the length, breadth of the outermost square and the diameter of the second circle is equal to 10, the area on square ABCD is 10×10=100

If you make a diagonal line between corners through the shaded square, you'll make a triangle. Using the formula a^2+b^2=c^2 we can determine the diameter of the circle. If the shaded square is 25^2 units, the length (a) and width (b) are both 5. 5^2+5^2=50. You don't have to know the square root of 50 to solve this. The next square you do the same thing, except your length and width will be sqrt 50. 50+50=100. That would be your hypotenuse of the triangle you make in the second square. Sqrt of 100 is 10, this is your length and width for the largest square, area of 100.

L of diagonnal =dia. Of circle Dia. Of circle = side of outer square Hence ans=100

Shaded square area is 25 so one side legnth is 5. The diagonal of that sqre is like the hypotenuse of a right triangle so that is equal to /sqrt(25+25)/ = /sqrt(50).Its the side of the next sqr. In the same way the the diagonal of the second sqr is equal to the side of the sqr ABCD. And it is /sqrt(sqrt(50^2+50^2))/ and that = /sqrt 100 =10 so the area =10*10=100

just an idea . it seems that square as 1/4

Intuitively obvious.

Firstly, observe and take into account the obvious to solve this problem.

1) Area = 25 therefor 5 must be the side of the smallest square. 2) The diameter of the smallest circle = the hypotenuse of the smallest square as well as the side of medium square. 3) Using either s² + s² = S² or the properties of triangle with two equal sides i.e. the hypotenuse = side√2 one can start the solving the sides. 4) medium side = √2•25 ; therefore largest side is √2 multiplied by the medium side squared resulting in 10.

After deducing this, the last step in multiplying 10 by 10 to reveal the answer 100 units² as the largest square's area.

simple, ABCD is four time the area of red square

sides of red square is 5. its diagonal is the diameter of the 1st circle. so diameter =5 root(2). this diameter is the next square's sides.. similarly, the square's diagonals becomes circle's diameter. so diameter = 5 root(2) * root(2) = 10. finally area of ABCD is 10*10 = 100..

100 squr units Description: Red square's area is= 25 square units So length=5 So diagonal length=√(25+25)=√50 So the 1st circle diameter is=√50 So the 2nd squre length=√50 So diagonal length=√(50+50) =√100=10 So 2nd circle daiameter=10 S0 ABCD squre’s length=10 S0 ABCD squre’s area is =100

The diagonal of the red square is 5(2) ^1/2. hence the dia of next circle & side of next square will be 5(2)^1/2, then diagonal of next square is 5(2)^1/2 (2)^1/2=5 2=10. Hence area of ABCD is 10*10 i.e 100 Square units.

do right triangles of the two sides of the inner square such as 5 is opposite and 5 is adjacent and the center is the hypnoses of the triangle. The line then becomes the diameter of the circle so (5^2 + 5^2)^(1/2) = (50)^(1/2). The circle within the square has a diameter equal to length of the outer square's side and so on. You do this till the last outer circle becomes (100)^(1/2) or 10. 10 times 10 =100 = area of a square