SOLUTION: http://imgur.com/5GLK6fa Each side of the triangle is 8. Find the area of the shaded region. Thanks!

Question 937886: http://imgur.com/5GLK6fa
Each side of the triangle is 8. Find the area of the shaded region.
Thanks!
Answer by TimothyLamb(4379) (Show Source): You can put this solution on YOUR website!
I will give you the strategy, and you can do the math:
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all angles of the equilateral triangle (call it T), are 60 (180/3)
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draw a line from the top point of the triangle to the midpoint of the bottom edge
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find the long leg "a" of the resulting right triangle with 30-60-90 angles:
a/sin(60) = 8/sin(90)
a = sin(60)*8/sin(90)
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draw another line from the left point of T to the midpoint of the right edge:
this line forms another 30-60-90 but smaller right triangle inside the top-right of T:
use proportions to find the hypotenuse of this smaller right triangle:
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h/sin(90) = 4/sin(60)
h = sin(90)*4/sin(60)
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note that h is the radius of the center circle:
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find the area of the circle of radius h:
areaWhiteCircle = pi*rr
areaWhiteCircle = pi*hh
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the area of each moon-shape is 1/2 the area of a circle of radius 4, minus one lens-shaped sliver of the white circle:
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together, the three lens-shaped slivers of the white circle have an area equal to the white circle area, minus the area of triangle T. Let's call the total area of the three lens-shapes U:
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U = areaWhiteCircle - (area of triangle T)
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answer:
the shaded area = 3*(1/2)*pi*4*4 - U
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