A cirele is inscribed in a equilateral triangle of side 24 cm. What is the area (in cm$$^2$$) of a square inscribed in the circle?

According to a question, we draw the diagram is given below 

Side of triangle = a

then $$( AB)^2 = (AD)^2 + (BD)^2 $$

$$\Rightarrow a^2 = (\frac{a}{2})^2 + (AD)^2 $$

$$\Rightarrow AD =\frac {\sqrt {3}} {2} a $$

Since AD is median . So  centroid O divides each median in 2:1 ratio.

AO : OD = 2;1 

AO = 2x , OD=x

AD = 3x 

$$ \frac{\sqrt {3}{2}}a = 3x $$

$$x= \frac{\sqrt {3} }{6}a$$ 

OD = $$ \sqrt {3} {6} a $$

PQ = $$2 \times  radius  of  circle $$

     = $$2 \times \frac{\sqrt{3}} {6} a = \frac{\sqrt {3}} {3} a = \frac{a}{\sqrt{3}}$$

Now side of square = $$\frac {doagonal} {\sqrt{2}}$$

                             = $$ \frac{a} {\sqrt{3} \sqrt{2}}  = \frac {a} {\sqrt{6}}$$

Area of Square A = $$(side)^2   $$

                         = $$(\frac{a}{\sqrt{6}}) $$

                         = $$ \frac{a^2}{6} $$

                       =  $$\frac{24 \times 24} {6}$$

                      =  96 sqare unit Ans