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

• Nov 14th 2011, 08:05 AM
RichardGB
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
• Nov 14th 2011, 08:23 AM
e^(i*pi)
Re: Calculating the height of a cone given just the volume and the angle of the apex
Quote:

Originally Posted by RichardGB
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 $\displaystyle V = \dfrac{1}{3}bh$ which for a circular base is $\displaystyle V = \dfrac{\pi r^2h}{3}$

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

Sub this expression for $\displaystyle r$ into $\displaystyle V$

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

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

As ever, you should always check I've not cocked up somewhere (Giggle)