.
The area of the rhombus is = = 2ab (four congruent right-angled triangles).
The side length of the rhombus is L = .
The perimeter of the rhombus is 4L = .
The semi-perimeter of the rhombus is s = = .
The radius of the inscribed circle into the rhombus is r = = = .
Regarding the formula r = for the radius of inscribed circle into convex polygon, where A is the area of the polygon
and s is its semi-perimeter, see the lesson
Area of n-sided polygon circumscribed about a circle
in this site.
Thus the area of the inscribed circle into the rhombus is = .
Finally, the ratio , which is under the question, is
= = .
Solved.
Also, you have this free of charge online textbook on Geometry
GEOMETRY - YOUR ONLINE TEXTBOOK
in this site.
The referred lesson is the part of this online textbook under the topic "Area of polygons".