Colored area

How large is the area colored brown inside a square of side 6 cm if each of the four brown circular segments is from a circle with a radius of the length of the square's side? The length of the circular segments is equal to the length of the side of the square. The situation is shown in the picture on the right.

Correct answer:

S =  13.0444 cm²

Step-by-step explanation:

$$\begin{gathered} a=6\text{cm} \\ n=4 \\ r=a=6\text{cm} \\ t=a/2=6/2=3\text{cm} \\ {r}^2=h_1^2+t^2 \\ {h}_1=\sqrt{r^2-t^2}=\sqrt{6^2-3^2}=3\sqrt{3}\text{cm}\doteq 5.1962\text{cm} \\ \sin \alpha /2=t:r \\ \alpha =2\cdot \arcsin (t/r)=2\cdot \arcsin (3/6)\doteq 1.0472\text{rad} \\ {S}_1=\pi \cdot r^2\cdot \frac{\alpha }{2\pi }=3.1416\cdot 6^2\cdot \frac{1.0472}{2\cdot 3.1416}\doteq 18.8496{\text{cm}}^2 \\ {S}_2=\frac{a\cdot h_1}{2}=\frac{6\cdot 5.1962}{2}=9\sqrt{3}{\text{cm}}^2\doteq 15.5885{\text{cm}}^2 \\ {S}_3=S_1-S_2=18.8496-15.5885\doteq 3.2611{\text{cm}}^2 \\ S=n\cdot S_3=4\cdot 3.2611=13.0444{\text{cm}}^2 \end{gathered}$$

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