Hello, svetka!

Another approach . . .

A circle is inscribed in an equilateral triangle of side 8.

Find the area of the circle. Code:

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/ \
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/ * * * \
/* *\
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/ r\ /r \
/* \ / *\
/ * * * \
/ * | * \
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/ * |r * \
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: - - - - - - - - 8 - - - - - - - - :

The area of an equilateral triangle of side $\displaystyle x$ is: .$\displaystyle A \:=\:\frac{\sqrt{3}}{4}x^2$

We have $\displaystyle x = 8$, so: .$\displaystyle A \:=\:\frac{\sqrt{3}}{4}(8^2) \:=\:16\sqrt{3}$

Formula: .The area of a triangle is: .$\displaystyle A \:=\:\tfrac{1}{2}pr$

. . . . . . . where $\displaystyle p$ is the perimeter, and $\displaystyle r$ is the radius of the inscribed circle.

The perimeter of this triangle is 24, so we have: .$\displaystyle \tfrac{1}{2}(24)r \:=\:16\sqrt{3}$

Hence: .$\displaystyle 12r \:=\:16\sqrt{3} \quad\Rightarrow\quad r \:=\:\frac{4\sqrt{3}}{3}$

Now you can find the area of the circle . . .