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

1. ## 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

2. ## Re: Calculating the height of a cone given just the volume and the angle of the apex

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 $V = \dfrac{1}{3}bh$ 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: $h = \sqrt[3]{\dfrac{3V}{\pi \tan^2 \theta}}$

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