Squares on a Rhombus
From the Pythagorean Theorem, the diagonal of a 6-by-6 square is .
Half this length is .
key triangle is shaded in the drawing below.
Since the two short sides of the shaded triangle are known, and its
obtuse angle is 120° (45° + 30° + 45°), the Law of Cosines can be
used to find the length of the third side.
shaded triangle is the key to the solution.
Each short side equals .
Using the Law of Cosines with an angle of 120°, the long side of the triangle
So, the area of the square is 54 square units.