I think the problem here is that you define' 'r as the radius at the top of the gauge, but the 'r' variable as you used it in the formula for volume of the cone is the raidus where the top of the water is, not the top of the gauge. Let's define variables a little more explicitly:

Let:

h_g = depth of water in the conical gauge

r = radius at water level

V = volume of water in the gauuge

R = radius at opening of the gauge

H_g = height of the gauge

h_cyl = height of water in a cylindrical gauge

The volume of water in the conical gauge is , but r is function of h, where r = h(R/H). So the volums as a function of 'h' is:

In a cylindrical gauge this volume would come to a height of: From this we can set h_cyl as follows:

So given height 'h' (which you obtain from your pressure sensor) you can calculate the rain fall in inches with this equaton.