Question 1081669
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The diagonals of rhombus have length 4 and 6 inches and a circle is inscribed in it. 
Find the area of the region remaining in the rhombus not occupied by the circle. 
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<pre>
The diagonals divide the rhombus in 4 congruent right-angled triangles.

The side of the rhombus is {{{sqrt((4/2)^2 + (6/2)^2))}}} = {{{sqrt(4 + 9)}}} = {{{sqrt(13)}}}.

The area of each small right-angled triangle is equal to A = {{{(1/2)*(4/2)*(6/2)}}} = 3.

Using the side of the rhombus as the hypotenuse of the small triangle and the radius of the inscribed circle as the altitude, 
you can write the same area as

S = 3 = {{{(1/2)*sqrt(13)*r}}}.

It gives you r = {{{6/sqrt(13)}}}.

Then the area of the circle is {{{pi*r^2}}} = {{{pi*(36/13)}}} squared inches.

The area of the rhombus is 4 times the area of the small right-angled triangle, i.e. 4*3 = 12 squared inches.

Finally, the area under the question is {{{12-pi*(36/13)}}} square inches.
</pre>

Solved.