Payton colored a composite shape made up semicircles whose diameters are the sides of a square. The result is a shape given below. Find the area of the shaded region (Petals) in terms of x? Calculate the area using the dimension in 2nd figure?

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1 Answer
Jul 29, 2016

Area #=8x^2(pi-2)#

Explanation:

The area of half a petal is given by

#(pi r^2)/4-r^2/2 = r^2/4(pi-2)#

See the attached figure:
in light blue is represented the circle's quarter #(pi r^2)/4#
in yellow is represented the triangle's area #r^2/2#
in red is represented the difference #(pi r^2)/4-r^2/2#

(Note: the halfed petal is rotated #45^@#)

so the four petals have an area of

#8 r^2/4(pi-2)= 2r^2(pi-2)# but #r = 2x# then

Area #=8x^2(pi-2)#

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