Difference between revisions of "2012 AMC 10B Problems/Problem 16"
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| + | Three circles with radius 2 are mutually tangent. What is the total area of the circles and the region bounded by them? | ||
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| + | <math> \text{(A)}\ 10\pi+4\sqrt{3}\qquad\text{(B)}\ 13\pi-\sqrt{3}\qquad\text{(C)}\ 12\pi+\sqrt{3}\qquad\text{(D)}\ 10\pi+9\qquad\text{(E)}\ 13\pi </math> | ||
To determine the area of the figure, you can connect the centers of the circles to form an equilateral triangle of length <math>4</math>. Find the area of this triangle to include the figure formed in between the circles. This area is <math>4\sqrt{3}</math>. | To determine the area of the figure, you can connect the centers of the circles to form an equilateral triangle of length <math>4</math>. Find the area of this triangle to include the figure formed in between the circles. This area is <math>4\sqrt{3}</math>. | ||
Revision as of 21:21, 16 February 2013
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Three circles with radius 2 are mutually tangent. What is the total area of the circles and the region bounded by them?
To determine the area of the figure, you can connect the centers of the circles to form an equilateral triangle of length 
. Find the area of this triangle to include the figure formed in between the circles. This area is 
.
To find the area of the remaining sectors, notice that the sectors have a central angle of 300 because 60 degrees were "used up" for the triangle. The area of one sector is 
. Then this area is multiplied by three to find the total area of the sectors 
.
This result is added to area of the equilateral triangle to get a final answer of 
.
This means (A) is the right answer.
