I have a plastic cup with the dimensions:
Height = 7.89 cm
base radius = 2.10 cm
top Radius = 3.25 cm
With the formula, $$ vCup = \dfrac{\pi H}3(R^2+Rr+r^2) $$ I could calculate to fit 180mL in the cup.
Applying the same formula, $$ vCup = \dfrac{\pi 4.56}3(2.25^2+2.25 * 2.10+2.10^2) $$ I could fit 67.68mL of coffe in the cup.
Question: I would like a formula to calculate the radius according to height, that is, the coffee level, since I have the maximum and the minimum radii.
Why is it complicated to measure this in everyday life?
You can just set up the equation below
$$\frac{h}{H} = \frac{x-r}{R-r}$$
to obtain the radius, given height $h$
$$x = r+ \frac{h}{H}(R-r)$$
where $R$ and $r$ are the maximum and the minimum radii, respectively, and $H$ is the height of the cup.