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Let's say we have a rhombus with diagonals $a$ and $b$, which contains an inscribed circle. How can we find the area of that circle in terms of $a$ and $b$?

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You may be able to generalise from this: – Ian Coley Oct 16 '13 at 19:16
The wiki page for rhombus provides the answer, though it doesn't derive it. – David H Oct 16 '13 at 19:20
up vote 3 down vote accepted

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Obviously, the radius of inscribed circle is also a height $h=OH$ of the right triangle $\triangle AOB$. To find it, use equations for triangle's area $$ S_{\triangle AOB} = \frac 12 \frac a2 \frac b2 = \frac {ab}8 = \frac 12 ch $$ where $c = AB$ is a hypotenuse. So $r = h = \frac {ab}{4c} = \frac {ab}{4\sqrt{\frac {a^2}4+\frac {b^2}4}} = \frac {ab}{2\sqrt{a^2+b^2}}$, and therefore $$ S_c = \pi r^2 = \frac {\pi a^2 b^2}{4\left (a^2+b^2 \right )} $$

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A another approach could be remembering the area formula for tangential quadrilateral which in this case would be: $Area = 2r*side$ and you can calculate area and the side in terms of $a$ and $b$. – Sawarnik Nov 17 '13 at 5:19

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