Question 118675
Cake batter volume is the real variable that you're interested in here. 
With a larger pan, you need more batter.
With a smaller pan, you need less batter.
For the same volume, you will have a definite relationship between the height and diamter.  
The volume of the cake batter depends on the diameter of the pan and on the height of the pan. 
For a cylindrical pan, the volume is
{{{V=(pi)*D^2/4*h}}}
Originally let's use the subscript 1 to refer to 8 inch diameter pan.
{{{V[1]=(pi)*D[1]^2/4*h[1]}}}
When it's a 12 inch diameter pan let's use subscript 2.
{{{V[2]=(pi)*D[2]^2/4*h[2]}}}
Now let's plug in some numbers.
{{{V[1]=(pi)*8^2/4*h[1]}}}
{{{V[1]=(pi)*64/4*h[1]}}}
{{{V[1]=16*(pi)*h[1]}}}
And for pan 2,
{{{V[2]=(pi)*D[2]^2/4*h[2]}}}
{{{V[2]=(pi)*12^2/4*h[2]}}}
{{{V[2]=36*(pi)*h[2]}}}
Also, remember the depth is the same. 
{{{h[1]=h[2]=h}}}
1.{{{V[1]=16*(pi)*h}}}
2.{{{V[2]=36*(pi)*h}}}
of re-writing the equation 2.
{{{h=V[2]/(36*(pi))}}}
Now plug that value back into equation 1.
1.{{{V[1]=16*(pi)*h}}}
{{{V[1]=16*(pi)*V[2]/(36*(pi))}}}
{{{V[2]=(36/16)*V[1]}}}
{{{V[2]=2.25*V[1]}}}
So you need 2.25 times as much batter, so you multply all of the batter ingredients by 2.25.