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.
An equilateral triangle means:
It has 3 equal sides and 3 equal angles of 60 degrees each. Since each angle of the larger square ==90 degrees, then the 2 smaller angles on either side==[90 - 60] / 2 ==15 degrees.
Now, you have an Obtuse isosceles triangle on each side of the larger square with angles of 15, 15 and 150 degrees with two equal sides of 2 units each.
Using the Law of Cosines, we therefore find the 3rd side of Obtuse Isosceles triangle to be:
c==Sqrt[2^2+2^2 - (2*2*2*cos(150)) ]
c==~3.864 units - which is also the side of the larger square.