Flakes

We describe a circle of the square, and we describe a semicircle above each side of the square. This created 4 flakes.
Which is bigger: the area of the central square, or the area of four flakes?

Correct answer:

S₁ =  1
S₂ =  1

Step-by-step explanation:

$$\begin{gathered} a=1 \\ {S}_1=a^2=1^2=1 \end{gathered}$$

$$\begin{gathered} r_1=a\mathrm{/}2=1\mathrm{/}2={\displaystyle \frac{1}{2}}=0.5 \\ {r}_2=\sqrt{2}\mathrm{/}2\cdot a=\sqrt{2}\mathrm{/}2\cdot 1\doteq 0.7071 \\ {S}_3=4\pi \cdot r_1^2\mathrm{/}2=4\cdot 3.1416\cdot 0.5^2\mathrm{/}2\doteq 1.5708 \\ {S}_4=\pi \cdot r_2^2=3.1416\cdot 0.7071^2\doteq 1.5708 \\ {S}_2=S_3-S_4+S_1=1.5708-1.5708+1=1 \end{gathered}$$

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