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Geometry Difficulty 4.6 AIME Prove it North Macedonia

In the square ABCDABCD with side length 4 cm4\,\mathrm{cm}, from each vertex arcs are circumscribed with radii that equal half of the side length of the square, like in the picture. Calculate the perimeter and area of the marked part of the square.

Figure 1

Solution

The four arcs form a circumference with radius r=2 cmr = 2\,\mathrm{cm}. We have that L=2rπ=4π cmL = 2r\pi = 4\pi\,\mathrm{cm} and
P=42−r2π=16−4π=4(4−π) cm2 P = 4^2 - r^2\pi = 16 - 4\pi = 4(4 - \pi)\,\mathrm{cm}^2

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