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−  == Problem ==
 +  #REDIRECT[[2003 AMC 12A Problems/Problem 15]] 
−  A semicircle of diameter <math>1</math> sits at the top of a semicircle of diameter <math>2</math>, as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a ''lune''. Determine the area of this lune.
 
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−  [[Image:2003amc10a19.gif]]
 
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−  <math> \mathrm{(A) \ } \frac{1}{6}\pi\frac{\sqrt{3}}{4}\qquad \mathrm{(B) \ } \frac{\sqrt{3}}{4}\frac{1}{12}\pi\qquad \mathrm{(C) \ } \frac{\sqrt{3}}{4}\frac{1}{24}\pi\qquad \mathrm{(D) \ } \frac{\sqrt{3}}{4}+\frac{1}{24}\pi\qquad \mathrm{(E) \ } \frac{\sqrt{3}}{4}+\frac{1}{12}\pi </math>
 
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−  == Solution ==
 
−  [[Image:2003amc10a19solution.gif]]  
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−  The shaded area is equal to the area of the smaller semicircle minus the area of a sector of the larger circle plus the area of a triangle formed by two radii of the larger semicircle and the diameter of the smaller semicircle.
 
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−  The area of the smaller semicircle is <math>\frac{1}{2}\pi\cdot(\frac{1}{2})^{2}=\frac{1}{8}\pi</math>.
 
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−  Since the radius of the larger semicircle is equal to the diameter of the smaller semicircle, the triangle is an equilateral triangle and the sector measures <math>60^\circ</math>.
 
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−  The area of the <math>60^\circ</math> sector of the larger semicircle is <math>\frac{60}{360}\pi\cdot(\frac{2}{2})^{2}=\frac{1}{6}\pi</math>.
 
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−  The area of the triangle is <math>\frac{1^{2}\sqrt{3}}{4}=\frac{\sqrt{3}}{4}</math>
 
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−  So the shaded area is <math>\frac{1}{8}\pi\frac{1}{6}\pi+\frac{\sqrt{3}}{4}=\frac{\sqrt{3}}{4}\frac{1}{24}\pi \Rightarrow C</math>
 
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−  == See Also ==
 
−  {{AMC10 boxyear=2003ab=Anumb=18numa=20}}
 
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−  [[Category:Introductory Geometry Problems]]
 