To this equilateral triangle are attached 3 congruent circle segments. The equilateral triangle with one attached segment (painted grey) forms one sixth of a circle with radius 10. Therefore the area of the complete shape is:
Two adjacent sides of the square are tangents to adjacent arcs(with radius 10) while the opposite sides of the square pass through vertices of the curved triangle. The distance between a vertex of the curved triangle to the corresponding tangent side of the square must be a radius too. Therefore the square has a side-length of r.