GMAT Geometry Problem

Geometry Level 1

In the diagram above, all the points are collinear (belong to the same line). The numbers indicate how many units each point away from zero. All the circles pass through point zero. The points 1 till 6 are the centers of the six circles. The area of the green region is $C\pi$ . What is $C$ ?

Problem Courtesy: Mike Mcgarry.

12 21 24 42 48

1 solution

Hana Wehbi
Jul 8, 2016

Relevant wiki: Length and Area - Composite Figures

if we look at the green part between circles $10$ , which has a radius of $5$ and $12$ , which has a radius of $6$ ,the area of that part is going to be: $25\pi-36 \pi= 11\pi$ .

Take the green part between points $6$ and $8$ , these circles have radii of $3$ and $4$ respectively, so the area of the green "lobe" in between these circles is $16\pi-9\pi= 7\pi$ .

Finally, we are going to consider the green "lobe" between points $4$ and $2$ . These circles have radii of $2$ and $1$ $\implies$ the area of the green "lobe" in between these circles is $4\pi-\pi=3\pi$ .

The total area of the green region is: $(11+7+3)\pi= 21\pi$ , $\implies \boxed{C=21}$

Background Area of Circle= pi*(radius)^2 In simple terms, there are 6 circles Area of Circle 1=pi Area of Circle 2=4pi Green area between Circle 2 and 1 is 4pi-pi=3pi Area of Circle 3=9pi Area of Circle 4=16pi Green area between Circle 4 and 3 is 16pi-9pi=7pi Area of Circle 5=25pi Area of Circle 6=36pi Green area between Circle 6 and 5 is 36pi-25pi=11pi Total Green Area is 3pi+7pi+11pi=21pi

Onyekachi Maduabuchi - 3 years, 9 months ago

Nice. Thank you.

Hana Wehbi - 3 years, 9 months ago

GMAT Geometry Problem

Problem Courtesy: Mike Mcgarry.

1 solution

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