U can also solve it using  as the variable.(see figure)

'r' being the radius of the given semicircle,

Let 'l' and 'b' be the length and breadth of the rectangle.

So,l=2rcos

b=rsin

 

so, area A=2r²sincos

               =r²sin2

 

dA/d= r²cos2x2

 

for maximum area,

dA/d=0

 or,r²cos2x2

i.e,cos2=0

or,=/4                            ... 0<</2

 

putting the value of   in the length and breadth ,u get the dimensions of the rectangle

l=2xr

b=r/2

so,

A=r²