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

  1. #1
    manic
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    [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
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  2. #2
    Super Member
    earboth's Avatar
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    Quote Originally Posted by manic View Post
    ... 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
    Attached Thumbnails Attached Thumbnails [SOLVED] Creating a frustum cone from a frustum cone-wasserhoehe_keglstumpf.gif  
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  3. #3
    Newbie
    Joined
    Sep 2008
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    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 ?
    Attached Thumbnails Attached Thumbnails [SOLVED] Creating a frustum cone from a frustum cone-cone.bmp  
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