# Math Help - [SOLVED] Creating a frustum cone from a frustum cone

1. ## [SOLVED] Creating a frustum cone from a frustum cone

hi, Im trying to calculate the volume of a frustum of a frustum of a cone. I understand how to calculate the volume of a frustum cone, but i need to make a frustum of the frustum cone( if that makes sense) Basically, if you imagine a frustum of a cone shaped vessel, that is partially full of liquid, im trying to calculate the volume of liquid, given;

top radius ( Rt )
bottom radius ( Rb )
height of vessel ( H )
height of fluid in vessel (measured from bottom). ( h )

I haven't done any trig for many years, and the best i could come up with was calculating the new top radius given the height of the fluid, then calculating the new volume should be easy. So far i have:

new top radius = Rb + (2 ( (H - h) * (Tan(arcTan((H / ((Rb - Rt) / 2)))))))

and the new top radius is then used to calculate the new frustum cone volume.

is there an easier way? it just seems a bit over complex to me

2. Originally Posted by manic
... Basically, if you imagine a frustum of a cone shaped vessel, that is partially full of liquid, im trying to calculate the volume of liquid, given;

top radius ( Rt )
bottom radius ( Rb )
height of vessel ( H )
height of fluid in vessel (measured from bottom). ( h )

I haven't done any trig for many years, ...
is there an easier way? it just seems a bit over complex to me
Maybe I've found an easier way to calculate the new radius.

I've attached a sketch of the situation.

Use similar triangles. To be accurate use similar right triangles. You can setup the following proportions:
1)

$\frac{R_t}{R_b} = \frac{x}{x+H}~\iff~ x=\frac{H \cdot R_t}{R_b - R_t}$ ....... Since all values of the RHS are known you can calculate the length of x.

2)

$\frac{R_n}{R_b} = \frac{x+(H-h)}{x+H}$ ....... Plug in the value of x from 1) and solve for $R_n$

3. ## Cone partial full

I have the same problem to calculate the volume but my cone is rotated 90 degrees. The height of the level of water inside the cone is known but how can i calculate the volume of liquid inside the cone ?