An equilateral triangle is constructed on each side of a square with side length 2 as shown below. The four outer vertices are then joined to form a large square. Find the side length of the large square.

[asy]

unitsize(1 cm);

draw((1,1)--(1,-1)--(-1,-1)--(-1,1)--cycle);

draw((1,1)--((1,1) + 2*dir(120))--(-1,1));

draw(rotate(90)*((1,1)--((1,1) + 2*dir(120))--(-1,1)));

draw(rotate(180)*((1,1)--((1,1) + 2*dir(120))--(-1,1)));

draw(rotate(270)*((1,1)--((1,1) + 2*dir(120))--(-1,1)));

draw((1 + sqrt(3))*dir(0)--(1 + sqrt(3))*dir(90)--(1 + sqrt(3))*dir(180)--(1 + sqrt(3))*dir(270)--cycle);

label("$2$", (0,1), S);

[/asy]

Guest Apr 8, 2023

#1**0 **

The equilateral triangle has side length sqrt(3), so the four outer vertices of the large square are sqrt(3) units away from each other. The large square has an inscribed circle, which is tangent to all four sides of the square. The radius of the inscribed circle is 2/sqrt(3), so the diameter of the inscribed circle is 2*sqrt(3). The side length of the large square is equal to the diameter of the inscribed circle plus the distance between the outer vertices, which is sqrt(3) + 2.

Guest Apr 8, 2023