Blue Area vs. Yellow Area

The ratio of a circle to the square that envelops it is the same regardless of the size. Multiplying it by 16 doesn't change the ratio, and as long as the squares are the same size and the circles all fit exactly, then the ratio is the same.

Let the side of the square be equal to $x$

The radius of the big circle comes to be $\frac{x}{2}$ and hence the white area comes to be

$\pi { \left( \frac { x }{ 2 } \right) }^{ 2 }=\boxed {\pi \frac { { x }^{ 2 } }{ 4 }}$

In the second square the diameter of each circle would be $\frac{x}{4}$ and hence the radius would be $\frac{x}{8}$ . Thus the area of the smaller circles would be

$\pi { \left( \frac { x }{ 8 } \right) }^{ 2 }=\quad \pi \frac { { x }^{ 2 } }{ 64 }$ . Since there are $16$ circles, the total white area would be

$\pi \frac { { x }^{ 2 } }{ 64 } \times 16\quad =\quad \boxed {\pi \frac { { x }^{ 2 } }{ 4 } }$ .

Since the white area in both the squares is the same, the shaded areas would also be same (obviously :P)

big circle radius = r small circle radius = 1/4 r

A(yellow)=(pi)(r^2)

A(blue)=(16)(pi)(r/4)^2 =(16)(pi)(r^2)/(16) =(pi)(r^2)

If we divide the red shaded square in 16 congruent squares such that each square contains exactly one circle.

Now consider the blue shaded square and one of the 16 congruent squares. In the above considered squares,we can directly state, without any calculation, that the ratio of shaded area and unshaded area in both is equal because both are squares in which the circle touches the boundary of the respective square.

Likewise we have the ratio of shaded area and unshaded area equal in all the 17 squares(16 red shaded+1 blue shaded).

So we can have, by properties of ratio and proportion, that, ratio of shaded area and unshaded area equal in both the larger red square and blue square and hence equal shaded area area as total areas of squares are same.

Just went with the logic that the shaded area that surrounds each circle in the red square is a percentage of the shaded area that surrounds the big circle in the blue square.

Let the radius of the small circles be $x$ . Then the area of one small circle is $πx^2$ . The area of $16$ small circles is $16πx^2$ . The length of a side of the square is $8x$ The area of the square is $64x^2$

Assume the radius of one small circle is $1$ unit. Then the difference between the areas gives the area of the red region which is is $64 - 16π$ or $16(4 -π)$

The radius of the large circle is $4x$ The area of the large circle is $16πx^2$ At this point we know the area of the large circle is equal to the area of $16$ small circles.

The difference between the area of square and the area of the large circle gives the area of the blue region which is also $64 - 16π$ or $16(4 - π)$

So the area of the red region is equal to the area of the blue region.

Let's say the the big circle has the radius of 1 and area of Pi. Then the square is 4 cm^2. The small circles have the quarter of the radius of the big circle. Therefore, the area of a small circle is 1/16*Pi. Since there are 16 circles, the area of small circles is Pi. Therefore, the shaded areas are equal.

let's pretend we live in a world where Pi = 1, it is a ratio after all and we'll give the edges of the square a value: 8 meaning for the big square r^2Pi=16 and each small square r^2Pi = 1 you can see that there are 16 small squares and their areas are 1 so the total area is 16, thus, they are equal!

Let Side Length of both Squares be "8a"

Then Radius of Large Circle in left with Yellow Region will be = 4a

So, Area of Yellow Region $\ce{A1}$ = Area of Square - Area of Circle = ( 8a x 8a ) - ( π x $\ce{4a^2}$ )

= (64 - 16π) $\ce{a^2}$

Radius of every Small Circle in right with Blue Region = a

So, Area of Blue Region $\ce{A2}$ = Area of Square - ( 16 x Area of Circle ) = ( 8a x 8a ) - { 16 x ( π x $\ce{a^2}$ )}

= (64 - 16π) $\ce{a^2}$

$\ce{A1}$ = $\ce{A2}$ i.e., Both have equal area

Let the radius of 1 small circle be r. The radius of the large circle is 4r. Total area of 16 little circles: 16 × (πr²) = 16πr² Total area of 1 big circle: π(4r)² = 4² × (πr²) = 16πr² 16πr² = 16πr² Thus, they have the same area.

Lets' assign one unit to the radii of the 16 small circles. 1) The side of the circumscribing square is 8, si its area is 64 .
2) The total area of the 16 circles is 16π . 3) The blue area is the area of the square minus the total area of the 16 circles, i.e. 64 - 16π . 4) The radius of circumscribed circle at the letft is 4 (half of that ot the square), so its area is π4^2 = 16π . 5) The yellow area is the area of the square minus the area of the circumscribed circle, i.e. 64 - 16π , which is also the yellow area.

Let R be the radius of the big circle. Let r be the radius of a small circle.

R = 4r.

Area of Big Circle => Pi * R^2 => Pi * 16 * r^2.

Area of the 16 small circles => 16 * Pi * r^ 2.

Both are the same thing, both subtract the same area from the square.

Let the radius of the big circle be r. Then, the area of the big circle is πr². The square around the big circle has the dimensions 2r x 2r, for an area of 4r². Then the area of the square minus that of the circle is 4r²-πr²=r²(4-π). The smaller circles have a radius of r/4. So their area is πr²/16 each. Since there are 16 of them, their total area is 16 x πr²/16, of πr². The blue region has the area of (2r x 2r)- πr². 4r²-πr²=r²(4-π)--which is the same as the yellow region. Thus, the regions have equal areas ☺☺☺☺

Yellow area, 4 shaded corners

Blue area, 64 shaded corners

The smaller circles have 1/4 the radius of he big one, so they each have 1/16th of it's area

1 yellow corner has the same area as 16 blue corners.

There are 4 yellow and 64 blue corners.

4*16=64

Same area.

Each square is the same size, so all we need to do is compare the white area in each square, to see which of the coloured areas is larger or if they are the same size.

If the radius of the big white circle in the left square is r , then said circle obviously has the familiar area of $\pi \times r^2$ .

Each of the small circles in the square on the right has radius of (1/4) r , which translates to an area of $\pi \times {(\frac{1}{4} r)}^2 = \pi \times \frac{1}{16} r^2$ per small circle. There are 16 of them, so the total area of all the white circles in the blue square is $16 \times \pi \times \frac{1}{16} r^2$ , which is just $\pi \times r^2$

In other words: the white areas in both squares have exactly the same size. Then what's left over must be the same size too. The answer to the question is thus $\large \color{#3D99F6}{\boxed{Equal \space area}}$ .

Left : 4 r2 - pi r2 right : 4 r2 - 16 pi(r/4)2

If the radius of the smaller circle is "r", then the area of that circle is pi (r^2). Thus, the addition of the 16 small circles of the right-side diagram is 16(pi (r^2)). The length of one side of the square that the circles are within is equal to the length of the diameter of the largest circle on the left-side diagram and the sum of four diameters (8 radii) of the smaller circles. This means that the radius of the largest circle is half of the length of one side of the square, or 2 diameters (4 radii) of the smaller circles. Letting "R" denote the radius of the largest circle, then R = 4r. The are of the largest circle is pi (R^2). substituting the "R" for 4r, the area of the largest circle is 16(pi (r^2)). This means that the are that the small circles occupy together and the large circle occupies are equal within two congruent squares, and the shaded area is thus the same area in both diagrams.

Considering the Radius of the First Circle as 4x Then the radius of Small Circle is x

then their area X number of circles

Pi(4x)^2 = 16x^2 Pi

Pi(x)^2 x 16 = 16 x^2 Pi

Theres no change so they are equal thus ,,

area of the same square minus

the same Area unshaded = The Same !

Let the side of the square be equal to x

The radius of the big circle comes to be x/2 and hence the white area comes to be.
π(x/2)² = π[x²/4]

In the second square the diameter of each circle would be x/4 and hence the radius would be x/8 . Thus the area of the smaller circles would be π(x/ 8)² = π(x/64)

. Since there are 16 circles, the total white area would be

π(x²/64)16 = πx²/4

Since the white area in both the squares is the same, the shaded area wold also be same (obviously :P). Thanks

Depends on: 1) the area of the square, 2) the size of the circle, 3) number of circles inside the square, 4) area of the circles and 5) the area the circles occupy. Since the first square has bigger circle than the other square, it doesn't mean that the area of the shaded region on the first square is bigger, so the answer is $equal$ .

This question correct answer is equal area

By observing the two pictures clearly we can say that the region of red is equal to the region of blue region, but may of us confuse and answered blue:)

We calculate areas of circles with in given squares first,witch is are equal in both cases.Therefore,area left out side with in both squares will be same.Ans.

It is very weird that we all think alike and most of us got the right answer its at times like this I understand how similar we all are

Assuming 'a' is the side of the square, Blue shaded area (aXa)- ( PiXaXa/4), and so is the red shaded area! That is simplify
(aXa) - 16(PiX(a/8)X(a/8) to get the same thing!

well I can't insert the symbol for 'Pi' :)

Both areas have the same size, cause the little circles anda the bigger circle used to ocup the same area in the square

consider the first firgure calc the relative area of the circle to that of the square which results to 78.5% and do the same for next figure, will get same results.

If you only look at one circle in the figure of many circles and imagine a square described of one circle, the ratio of only square area to the circle will not change. And since the two figures have the same total are the ratio will be the same.

so simple. as we can see that the small circles are in large numbers and they have cover more area than the others. but confusion is there that we think they are not shaded but actually they are shaded with white color.