This is a lot more complicated than it first seems..
I found this nifty website that calculates the volume of a truncated cone..
http://keisan.casio.com/has10/SpecExec.cgi
The site also gives the formulas used in their algorithm. I'm pretty confident that with this information we can find the "midpoint".
In your drawing you say that you have the circumferences of the the two circles that make up the top and bottom of the cup, but i'm assuming that you meant to say diameter because of the way you've drawn your dimensions. If you do actually mean circumference you can easily solve for the radiuses using the formulas:
Circumference = π*d and Radius = d/2
Anyway, back to the actual problem..
When you punch in the dimensions of your cup it stays that the volume is 46.63
Now divide that number by two and and set it equal to the volume formula they provide and sub in all of the information you know..
This gives us an equation that looks like this:
23.31 = (1/3)*π*(R1^2+ R1(1.25) + (1.25)^2)*H
Where:
23.31 is half of the total volume
R1 is the unknown radius of the circle that is the top of the new "half cup"
R2 is the radius of the bottom of the cup
and H is the unknown height.
Now it appears that we cannot solve this for H because we also have R1 as an additional unknown, making 2 unknowns and only one equation. However, because of the cups conical shape, we can relate R1 to H using the principle of similar triangles.
If you look back at the original drawing, we see that the slope of the side of the cup can be described by the difference in the size of the radiuses on it's ends and the total distance over which they change.
Slope = (1.87-1.25)/6
Slope = 0.1 (approximately)
So now that we know the slope we can say that the radius at the middle is just the height at the middle (H) times the slope (.1) plus the radius at the bottom
or written mathematically as..
R1 = .1*(H) + 1.25
Now we can actually solve for H!
Here's the equation I came up with the 2 unknowns..
23.31 = (1/3)*π*(R1^2+ R1(1.25) + (1.25)^2)*H
Now sub (.1*H+1.25) in anywhere R1 appears and we get..
23.31 = (1/3)*π*[(.1*H+1.25)^2 + (.1*H+1.25)*(1.25)+(1.25)^2]*H
^ This is literally a nightmare to try and solve so i'm just gonna use wolframalpha.com
http://www.wolframalpha.com/input/?i=solve+23.31+%3D+%281%2F3%29*%28pi%29*%5B%28.1*H%2B1.25%29%5E2+%2B+%28.1*H%2B1.25%29*%281.25%29%2B%281.25%29%5E2%5D*H+for+H
which says that H = 3.607
