The diameter of the liquid's surface is a function of the liquid's height. It will be

d = (20 cm)/(30 cm)*h = (2/3)h

Then, the volume will be

v = (1/3)*(π/4)d^2*h

= π/12*(2/3h)^2*h

= π/27*h^3

The derivative of this expression is

dv/dt = (π/27)*(3h^2)*dh/dt = π/9*h^2*dh/dt

Solving for dh/dt, we get

(dv/dt)*9/(π*h^2) = dh/dt

With your numbers, we have

(0.2 cm^3/s)*9/(π*(12 cm)^2) = dh/dt

(1.8/(144π)) cm/s = dh/dt

dh/dt ≈ 0.00398 cm/s

Perhaps the key is to realize that the diameter is a function of height. It is not constant. So, the volume is proportional to the cube of height. The cube introduces a factor of 3 to the derivative.

d = (20 cm)/(30 cm)*h = (2/3)h

Then, the volume will be

v = (1/3)*(π/4)d^2*h

= π/12*(2/3h)^2*h

= π/27*h^3

The derivative of this expression is

dv/dt = (π/27)*(3h^2)*dh/dt = π/9*h^2*dh/dt

Solving for dh/dt, we get

(dv/dt)*9/(π*h^2) = dh/dt

With your numbers, we have

(0.2 cm^3/s)*9/(π*(12 cm)^2) = dh/dt

(1.8/(144π)) cm/s = dh/dt

dh/dt ≈ 0.00398 cm/s

Perhaps the key is to realize that the diameter is a function of height. It is not constant. So, the volume is proportional to the cube of height. The cube introduces a factor of 3 to the derivative.