Question

A circle inscribed in an equilateral triangle.If the circumference of the circle is $176cm$, find the area of the shaded region ?

Answer 1

Given circumference of circle$=176$cm

$\Rightarrow 2\pi r=176\Rightarrow 2.\frac{22}{7}.r=176\Rightarrow r=28cm$

Here, $AE,BF$ and $CD$ are the medians of triangle and $O$ is the centriod of triangle such that $O$ divides $AE$ in the ratio

$AO:OE=2:1\Rightarrow \frac{AO}{OE}=\frac{2}{1}\Rightarrow AO=2.28\left(OEistheradius\right)\Rightarrow AO=56cm\therefore AE=AO+OE=56+28=84cm$

Here $OE$ is radius and $BC$ be tangent of circle than,

$OE\perp BCLet,AB=x\therefore in△AEBAB=\sqrt{{AE}^2+{BE}^2}\Rightarrow x=\sqrt{{\left(84\right)}^2+{\left(\frac{x}{2}\right)}^2}\Rightarrow x=56\sqrt{3}cm$

therefore area of shaded region$=\frac{1}{2}\times BC\times AE-\pi r^2=\frac{1}{2}.56\sqrt{3}.84-\frac{22}{7}.{28}^2=1510.88{cm}^2$