In the figure above, A and B are the centers of the two circles. If ea

I could not not understand the problem can anyone please explain it a bit more in detail?

GMATinsight

How do you know tha the Central angle is 120 degree?

Also if we know the know the central angle can we just find the area of the sector and multiply it to 2 to find the shaded region area?

Can someone provide another solution? I did not really understand the solution above. How can we determine the angle of the diamond shape inside the shaded area?

I could not understand why the angle is taken 120. Please help me with its solution.

took me a bit to Understand:


Step 1: Draw the 2 Intersection Points between the Circle and label them A and D, like in the picture above mine.


Step 2: Draw the following Radii = X:

-1- from Center - C1 of the Left Circle to Intersection Point A

-2- from Center -C1 of the Left Circle to Center- C2 of the Right Circle

-3- from Center - C1 of the Left Circle to Intersection Point D


if you label all the Radii = X, you will see that the Diamond Shape in the middle is composed of TWO Equilateral Triangles of Side = X



Step 3: Find Area of Shaded Region:


(1st) Focusing on the Sector Area of the Left Circle. The Central Angle created at Center - C1 with Line Segments X connected to Point A and Point D ----> is a 120 Degree Central Angle.

The Sector Area of this will include the Diamond in the Middle + the Outer Portions OUTSIDE the Diamond but INSIDE Circle A


2nd) Do the Same from the Perspective of the Right Circle. Again, you have the same 120 Degree Central Angle.

This Sector Area will include the Diamond AGAIN in the Middle + the Outer Portions OUTSIDE the Diamond but INSIDE Circle B



3rd) ADD these 2 Equivalent Sector Areas.

This will account for ALL of the Shaded Region. BUT --- we Actually included the DIAMOND AREA TWICE ---- we DOUBLE-COUNTED this Diamond Area


To make up for this Over-Counting, we need to Subtract ONE Area of the DIAMOND Region from the Addition.

Summary:

Area of Shaded Region = 2 * [Sector Area with 120 deg. Central Angle] - [Area of Diamond]




I. Sector Area with 120 deg. Central Angle. Radius = X

(120/360) * (pi) * (X)^2 = (1/3) * (pi) * (X)^2

again, we want to ADD Another to get the entire Region (with the Double-Counted Diamond)

2 * (1/3) * (pi) * (X)^2---- (equation 1)


II. Now, we need to SUBTRACT the Diamond Area to avoid the Double-Counting

the Diamond is composed of 2 Equilateral Triangles with Side = Radius = X

2 * [ (X)^2 * sqrt(3)] / 4 =

1/2 * (X)^2 * sqrt(3) ----- (equation 2)



III. Subtract (equation 2) FROM (equation 1) to get the Correct Area of the Shaded Region


[ 2 * (1/3) * (pi) * (X)^2 ] - [ 1/2 * (X)^2 * sqrt(3) ]


----LCD = 6-----


[(4/6) * (pi) * (X)^2 ] - [(3/6) * (X)^2 * sqrt(3) ]


----take (X)^2 as a Common Factor)

(X)^2 * [ (4 * (pi) - 3 * sqrt(3) / 6]


Answer -C-


I hope it didn't confuse anyone anymore and helped in some little way

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