Question

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?

Answer 1

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)`