The lateral area is pi*r*s where s is the slant height of the cone. So in this case pi*r*s=12pi so the slant height s = 12pi / 2pi = 6. The height of the cone will be h = sqrt(6^2 - 2^2) by Pythagoras or h=sqrt(32) = 4*sqrt(2). Now for the volume: for a cone V = (1/3)*pi*r^2*h so here V = (1/3)*pi*2^2*4*sqrt(2) which gives us V = (16/3)*pi*sqrt(2) or answer A.
