55 of 100: Rings

Since the circles are all congruent and the area of overlapping is same

Therefore total blue area=total pink area

So area of the total figure is 44

Now say area of one circle is x

So 4x-3×4=44 and x=14

Because the overlapping area are counted 3 times extra.

Obviously, the blue shaded area ( $22$ ) is half of the rings. Thus, we can find the area of these overlapped circles.

The area is just the area of the overlapped areas ( $4 \times 3 = 12$ ), subtracted from the area of the 4 circles; let us call it $4 \times A$ ). Thus, we get $4A-12$ .

Thus, we can write an equation: $\frac{4A-12}{2} = 22 \longrightarrow 4A-12=44 \longrightarrow \boxed{A=14 }$

The total area is $44$ . Now let the area of one circle be $C$ . Then notice also that the total area is $4C -3\times 4$ as we have four circles minus three intersections.

Equating and solving for $C$ gives $\boxed{14}$ as required.

Total area is 44 by symmetry. This is 4 circles minus three overlaps of area 4 each. Thus 4 circles is 56, leaving each to be 14

Same solution as @Kevin Tran 's

We note that the bottom left end part of the blue portion can flip to the top and fit into the purple part of the fourth circle, part of the third overlap and the third circle as shown in the bottom figure. Let the area of each circle be $A$ , then the blue area is:

$\begin{aligned} \text{Area of blue portion} & = \text{Area of two circles} - \text{An extra overlap (blue)} - \text{Half an overlap (purple)} \\ 22 & = 2A - 4 - 2 \\ A - 3 & = 11 \\ \implies A & = \boxed{14} \end{aligned}$

My approach was visual. Enumerate the circles from left to right, 1, 2, 3 and 4.

Constructing the blue part is equivalent to constructing the pink part, by symmetry. Thus the blue area is half the figure.

Looking at the blue part withing 1 and 2 that is not in 3, it fits into 3 and 4 except from the intersection of 2 and 3.

We know the intersection must be halved (if you fill up the two circles, the pink and the blue part must share the intersection evenly). We also know the intersection of 3 and 4 is overcounted once.

Call the area of the two circles X. Now 22 = X - 2 - 4 (Where the last four is for overcounting the intersection of 3 and 4, and the minus two is for the missing half intersection.)

So X = 28, which leaves one circle as 14.

The blue shaded figure is 22 units in area. Since it encompasses both the lowest point on the left-most circle and the highest point on the right-most circle, the line bisects the total area of the figure. Therefore, the total area of the figure would be 44 units. Then, the area of the overlaps is given as 4 units each. Thus the total area of the four circles, standing apart would be 44+12. This gives us 56 units total, divided among 4 circles, or a value of 14 for each independent circle. ☺☺☺☺

La figura es simétrica respecto de la recta que une A con B. Por lo tanto el A_total = 44. Si sumamos a este área las 3 superposiciones tenemos 44+12= 56 que corresponde al área de 4 círculos, por lo tanto:

Área de un círculo= 56/4= 14

Each circle comprises two of the overlap sections and one center section, so there are five overlaps and four centers in the overall piece.

Double 22 to find the total area is 44, subtract 20 for five overlaps to find the four centers equal 24, so each center is 6.

Center plus two overlaps = 6 + 2(4) = 14

Oooh, it was really painful... I invented the formula to calculate overlapping area. Then I wrote an auxiliary Python script to verify for given circle area whether is it possible to have such overlapping area and blue shaded area.
The closest result was for Area =14, so I picked 14. http://ideone.com/P4aRkZ

Ab = area shaded blue (= 22)

Ap = total pink area of the figure

Ao = area of overlap region (= 4)

Ac = area of one circle

Blue area Ab is half the area Ap of the total pink figure, to have the area of four circles whe need to had 3 times the overlap area Ao... so the area of a circle Ac is...

$A_{c} = \frac{2A_{b}+3A_{o}}{4} = \frac{44+12}{4} = \boxed{14}$

Total area of violet color is 2 x 22 = 44 ( Double of blue area)

Total area of isolated four circles = area of violet color (44) + 3 (Overlapped zones, Dark violet) = 44+ 3 x 4 = 56

Area of each Circle is = 56 / 4 = 14

The overall area , i.e the union of the 4 circles is equal to 2* Area of Shaded Region = 2*22 = 44

Let the area of each circle be x.

The area of the union of the 2 overlapping congruent circles = 2x - area of overlap

For n overlapping congruent circles we would have overall area = n x - (n-1) y where y is the area of the overlap.

So, for 4 overlapping circles, we have ,

Overall area = 4x - 3*4

44 = 4x - 12

4x = 56

x = 14

In this case, since the circles do not pass through the center of the other circles, finding the distance between the radii is not straightforward

The total colored area is $4\pi r^2-3×4$

Half of it is blue and half purple, by symmetry, so $2\pi r^2-3×2=22$ .

Dividing by 2 and adding 3 on both sides gives $A=\pi r^2=\boxed{14}$

As the circles are congruent, using symmetry (flipping the figure vertically),

$total\hspace{1mm} area\hspace{1mm} enclosed =2×\hspace{1mm}blue \hspace{1mm}region$ . $=>4×\hspace{1mm}circle$ $-$ $3×\hspace{1mm}common \hspace{1mm}area$ $=2×22$ $=> area\hspace{1mm} of \hspace{1mm}one \hspace{1mm}circle =\frac{44+3×4}{4} = 14$

Blue shade area is 22 which is half the area, so the entire area is 44. Going left to right, I will assume each circle is A,B,C,and D. Because of the overlap, you can give each circle half the overlap area. So that A is missing 2, B is missing 4, C missing 4, and D missing 2. So 44+2+4+4+2 reveals the total area of all four circles is 56. (56/4=14) So each circle's area is 14.

Let 'P' equal pink area, and 'B' equal blue area.

$P=B$

So, $P+B=44$

44 is equal to the 4 circles with only one set of overlaps, so $44+3•4=56$

56 is equal to the area of the four circles.

Then, $56/4=**14**$

The blue area is half of the full area, so we know the full area is 44. If we want to find the area of the four circles separated apart, then we would have to add back the 3 overlapping sections. Adding these back, we get 56 as the area of the four circles. Therefore, the area of one circle is 56/4 = 14.

P.S. Brilliant Staff - The problem is written the wrong way. It says that each pair of circles has overlapping area of four, but only each pair of circles which are next to each other satisfy this.

Let the required area be A, then 4A - 12 = 2(22), 4A = 56, A = 14.