# 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?

Jul 29, 2016

Area $= 8 {x}^{2} \left(\pi - 2\right)$

#### Explanation:

The area of half a petal is given by

$\frac{\pi {r}^{2}}{4} - {r}^{2} / 2 = {r}^{2} / 4 \left(\pi - 2\right)$

See the attached figure:
in light blue is represented the circle's quarter $\frac{\pi {r}^{2}}{4}$
in yellow is represented the triangle's area ${r}^{2} / 2$
in red is represented the difference $\frac{\pi {r}^{2}}{4} - {r}^{2} / 2$

(Note: the halfed petal is rotated ${45}^{\circ}$)

so the four petals have an area of

$8 {r}^{2} / 4 \left(\pi - 2\right) = 2 {r}^{2} \left(\pi - 2\right)$ but $r = 2 x$ then

Area $= 8 {x}^{2} \left(\pi - 2\right)$