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Math Help - Calculating the height of a cone given just the volume and the angle of the apex

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    Calculating the height of a cone given just the volume and the angle of the apex

    I think the header summarises what I would like to do.

    Regards
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    Re: Calculating the height of a cone given just the volume and the angle of the apex

    Quote Originally Posted by RichardGB View Post
    I think the header summarises what I would like to do.

    Regards
    What is the apex angle measured against? Is it a vertical line which falls from the apex perpendicular to the base? (ie: bisecting the cone in the Z axis)

    The volume of a cone is V = \dfrac{1}{3}bh which for a circular base is V = \dfrac{\pi r^2h}{3}

    We can also "draw" a triangle with the angle made with the apex being \theta. Since this is a right-angled triangle we can use trig: \tan \theta = \dfrac{r}{h} \Leftrightarrow r = h\tan \theta

    Sub this expression for r into V

    V = \dfrac{\pi (h \tan \theta)^2 h}{3} = \dfrac{\pi h^3 \tan^2 \theta}{3}

    Rearrange for h: h = \sqrt[3]{\dfrac{3V}{\pi \tan^2 \theta}}


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