The figure above shows three tangent circles (they touch at exactly on

Question

area.png The figure above shows three tangent circles (they touch at exactly one point), each with their centers along the same horizontal axis. The biggest circle contains two smaller ones. Of the two smaller circles, the larger one has a radius of 6 inches and the smaller one has a radius of 3 inches. What is the area of the shaded region? A. 45\pi B. 27\pi C. 24\pi D. 18\pi E. 9\pi

KarishmaB · Answer

Shaded region = Area of large semi circle - Sum of areas of two smaller semi circles

Sum of diameters of the two smaller circles = 6*2 + 3*2 = 18 = diameter of the large circle
Radius of large circle = 9
Area of large circle  = \pi*9^2 = 81\pi

Sum of areas of two smaller circles  = \pi*6^2 + \pi*3^2 = 45\pi

Shaded region  = 81\pi/2 - 45\pi/2 = 36\pi/2 = 18\pi

Answer (D)

DesiGmat · Answer

As the shaded region is in the Semi-Circle of the larger circle -

Diameter of the larger circle = Diamter of the two smaller circles = 2*6 + 2*3 = 18
Area of the larger circle = pi*9*9 = 81pi
Area of the larger semi circle = (81*pi)/2
Area of the 2 circles = 36*pi + 9*pi = 45*pi
But as mentioned above we are considering the semi circle
Hence area of the shaded region =(81*pi - 45*pi)/2 = 18*pi
Hope it helped.

EMPOWERgmatRichC · Answer

Hi All,

Since this question involves dealing with "half" of a picture, you can either do the "halv-ing" at multiple points in the question or do it at the end.

I'm going to call the circles "large", "medium" and "small"

The large circle has an area of (9^2)pi = 81pi
The medium circle has an area of (6^2)pi = 36pi
The small circle has an area of (3^2)pi = 9pi

Subtracting the medium and small circle from the large circle gives us...

81pi - 36pi - 9pi = 36pi

36pi is the area of everything in the large circle that is NOT taken up by the medium and small circles. Since the question asks for the shaded region, we know that the shaded region is HALF of THAT 36pi.

36pi/2 = 18pi

Final Answer:  D

GMAT assassins aren't born, they're made,
Rich

vimoha · Answer

Hi i did not understand why we need to divide the area 36 pie  by 2  , can somebody explain.

EMPOWERgmatRichC · Answer

HI vimoha,

If you start with the drawing, you'll see the large circle with the two smaller circles within it (all of the circles are tangent to one another and their widths from the DIAMETER of the larger circle).

Once we have the area of the larger circle, we subtract the areas of the two smaller circles - that leaves us with the 'leftover area' -'2 chunks' in the big circle - the 'top section' and the 'bottom section.' Those two chunks have the SAME area though. Since the question asks for JUST the SHADED area, we have to divide that 'leftover area' by 2.

GMAT assassins aren't born, they're made,
Rich

The figure above shows three tangent circles (they touch at exactly on

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