The area of the equilateral triangle is equal to (((√(16))(√8))/(3(√π))) Calculate the area of the circle inscribed in the triangle.


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Question Number 72997 by yannickmendes_33 last updated on 05/Nov/19

$T h e a r e a o f t h e e q u i l a t e r a l t r i a n g l e i s e q u a l t o \frac{\sqrt{16} \sqrt{8}}{3 \sqrt{π}}$ $C a l c u l a t e t h e a r e a o f t h e c i r c l e i n s c r i b e d i n t h e t r i a n g l e .$
	Answered by Kunal12588 last updated on 05/Nov/19

	$a r e a o f e q u i l a t e r a l △ = \frac{\sqrt{3} a^{2}}{4} = \frac{\sqrt{16} \sqrt{8}}{3 \sqrt{π}} = \frac{8 \sqrt{2}}{3 \sqrt{π}}$ $\Rightarrow \frac{a^{2}}{4} = \frac{8 \sqrt{2}}{3 \sqrt{3 π}}$ $a r e a o f c i r c l e = π r^{2}$ $\frac{h_{1}}{h_{2}} = \frac{2}{1}$ $\Rightarrow h_{1} = 2 h_{2}$ $h_{1} + h_{2} = h$ $\Rightarrow 3 h_{2} = h$ $\Rightarrow h_{2} = \frac{h}{3}$ $h_{2} = r = \frac{h}{3}$ $a r c i r c l e = π r^{2}$ $= π {(\frac{h}{3})}^{2}$ $= π {(\frac{1}{3} \times \frac{a \sqrt{3}}{2})}^{2}$ $= π \times \frac{1}{3} \times \frac{a^{2}}{4}$ $= π \times \frac{1}{3} \times \frac{8 \sqrt{2}}{3 \sqrt{3 π}}$ $= \frac{8 \sqrt{6 π}}{27}$
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