Math experts: What is the relationship between adding constant increments of pancake batter and the radius of the pancake?

The relationship is volume of pancake batter to surface area of pancake in pan. The assumption is that the batter always spreads out at a consistent rate to a set thickness.

so if a ¼ cup results in pi*r^2 size pancake, then ½ cup results in 2 (pi*r^2); ¾ cup batter equals 3(pi*r^2) size pancake.

A = π r^2

A / π = r^2

√ ( A / π ) = r

A is the area of the pancake, which (assuming uniform thickness) is directly proportional to the amount of batter you add,

Therefore, the radius of the pancakes will be proportional to the square root of the quantity ( Batter_used divided by Pi )

In this relationship, your equation would need three dimensions if you start with an initial volume because then technically your pancake can be infinitely thin with an infinitely large radius. So, use @jerv‘s equation, but add height h:

√(A/hπ) = r

A is your independent variable, h is a parameter, and r is the dependent variable, the radius.
@jerv The relationship is not proportional; as @gorillapaws pointed out, the radius increments get smaller and smaller the bigger the pancake gets.

@dxs I think there has been a misunderstanding.

1) I specifically stipulated a uniform thickness, though I did leave it unbounded with the implication that common sense would prevail and put it between zero and infinity; I have yet to even hear of any object of either infinite or zero thickness.

2) Where did I say Area was directly proportional to batter? I think you missed the part where I said, ”...square root of the quantity…”. But to rephrase;

( 1.732 – 1.414 ) < ( 1.414 – 1 ) yet ( 3 – 2 ) = ( 2 – 1 )

Watch those exponents ;)

Doesn’t it matter how hot the pan is?