SOLUTION: The 4-leaf figure is obtained by drawing semicircles on each side of a square with side 8. Find the area of the shaded region. Picture for more detail: http://imgur.com/dFgevEi

Question 937864: The 4-leaf figure is obtained by drawing semicircles on each side of a square with side 8. Find the area of the shaded region.
Picture for more detail: http://imgur.com/dFgevEi
Thanks!
Found 2 solutions by TimothyLamb, MathLover1:
Answer by TimothyLamb(4379) (Show Source): You can put this solution on YOUR website!
find the length of the hypotenuse of the isosceles triangle formed by the center of one circle and the two corners of one of the leaves:
---
h = sqrt( aa + aa )
h = sqrt( 4*4 + 4*4 )
h = 5.65685424949
---
find the area of one complete circle:
circleArea = pi*rr
circleArea = pi*4*4
circleArea = 50.2654824574
---
find the area of the square formed by the corners of the two right-most leaves and their reflection about the vertical right edge of the 8x8 square:
squareArea = hh
squareArea = 5.65685424949*5.65685424949
squareArea = 32
---
find the area of two leafs:
---
areaTwoLeafs = circleArea - squareArea
areaTwoLeafs = 50.2654824574 - 32
areaTwoLeafs = 18.2654824574
---
find the area of four leafs:
---
areaFourLeafs = areaTwoLeafs * 2
areaFourLeafs = 18.2654824574 * 2
areaFourLeafs = 36.5309649148
---
answer:
shaded area = 36.5309649148 sq.units
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Answer by MathLover1(20850) (Show Source): You can put this solution on YOUR website!
The square has area .
the semi circle radius is
The top and bottom white region together have area minus times the area of the semi circle, that is,

=> since it will be

=>
=>
=>
I will use this image because I cannot do shaded areas here:

http://imgur.com/Dpe1rJR

as you can see on a picture, radius is (disregard it), our radius is

The area of the quarter circle is

and the area of the triangle is

Thus, the area of the half-leaf (in red) is

There are two per leaf and the area of one leaf is

therefore the area of four leaves is:

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