Refer to figures below
From the given data we have
#D_1=6 ft.# => #r_1=3ft.#
#D_2=4 fl.# => #r_2=2ft.#
#h=12ft.#
As we can see in Fig. (a)
#r_2/(H-h)=r_1/H#
#2/(H-12)=3/H# => #2H=3H-36# => #H=36#
Since #s=sqrt(2)*r# (see Fig.(b))
#s_1=3sqrt(2)#
#s_2=2sqrt(2#
Area of bases
#S_(b1)=s_1^2=(3sqrt(2))^2=18#
#S_(b2)=s_2^2=(2sqrt(2))^2=8#
Side of the slanted trapezoid (see Fig.(c))
#x^2=h^2+(r_1-r_2)^2#
#x=sqrt(12^2+1^2)=sqrt(145)#
Height of the slanted trapezoid (see Fig.(d))
#y^2=x^2-((s_1-s_2)/2)^2#
#y^2=(sqrt (145))^2-((3sqrt(2)-2sqrt(2))/2)^2#
#y^2=145-1/2=289/2# => #y=(17sqrt(2))/2#
Area of each slanted trapezoid
#S_(trapezoid)=(s_1+s_2)/2*y=(3sqrt(2)+2sqrt(2))/2*(17sqrt(2))/2# => #S_(trapezoid)=85/2#
Total Area
#S_T=S_(b1)+S_(b_2)+4*S_(trapezoid)=18+8+4*85/2=196 ft.^2#
