External squares are drawn on each side of a rhombus, and the centers of these squares are joined to form convex quadadrilateral Q. If the length of each side of the rhombus is 6, and if one of its angles has a degree-measure of 30, find the area of the quadrilateral Q.
I have not done a full answer, and you will need to checke what I have done very carefully.
Here is my outline.
There are lots of angles you can work out on this diagram. You were given 30 degrees as one of the rhombus angles, so the others must be 30, 150,150.
Since the sides of all the given squares is 6.
The distance from a vertex to the centre of its square is 3 sqrt2 (pythagoras's theorem)
All the green triangles are congruent.
Therefore the new quadrilateral is a square.
Using cosine rule I got side squared = 54 units squared.