Question 1085018
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The area of the rhombus is   {{{A[rhombus]}}} = {{{4*((ab)/2)}}} = 2ab     (four congruent right-angled triangles).


The side length of the rhombus is   L = {{{sqrt(a^2+b^2)}}}.


The perimeter of the rhombus is   4L = {{{4*sqrt(a^2 + b^2)}}}.


The semi-perimeter of the rhombus is   s = {{{(4L)/2}}} = {{{2*sqrt(a^2 + b^2)}}}.


The radius of the inscribed circle into the rhombus is   r = {{{A[rhombus]/s}}} = {{{(2ab)/(2*sqrt(a^2+ b^2))}}} = {{{(ab)/sqrt(a^2+b^2)}}}. 


<pre>
     Regarding the formula  r = {{{A/s}}}  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 
         <A HREF=https://www.algebra.com/algebra/homework/Surface-area/Area-of-n-sided-polygon-circumscribed--about-a-circle.lesson>Area of n-sided polygon circumscribed about a circle</A>
     in this site.
</pre>

Thus the area of the inscribed circle into the rhombus is &nbsp;{{{A[circle]}}} = {{{pi*((a^2*b^2)/(a^2 + b^2))}}}.


Finally, &nbsp;the ratio &nbsp;{{{A[circle]/A[rhombus]}}}, &nbsp;which is under the question, &nbsp;is


{{{A[circle]/A[rhombus]}}} = {{{(pi*((a^2*b^2)/(a^2 + b^2)))/(2ab)}}} = {{{(pi*a*b)/(2*(a^2+b^2))}}}.



Solved.



Also, &nbsp;you have this free of charge online textbook on Geometry

&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson>GEOMETRY - YOUR ONLINE TEXTBOOK</A> 

in this site.



The referred lesson is the part of this online textbook under the topic "<U>Area of polygons</U>".