The volume of a frustum of a cone

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Question from CV:

If I know the height, taper, and volume of a frustum cone,
what are its Radii?

Calculating frustum cone volume is straight forward.
v=Pi/3*h*(R*R+r*r+R*r) where;
v=volume;
h=frustum height;
R=major radius;
r=minor radius;
Pi=3.1415926;
t=taper, ('slant angle' where t=0 is a disk)

Here is the tricky part.
Knowing 'h', 't', & 'v';
Calculate one or both of 'R' & 'r'.

Specifically, what is the formula for 'R=' or 'r='?

This is MUCH harder than you may think.

Thanks,

Hi,

I drew a diagram of the frustum of a cone and added some lines and labels.

cone

From the triangle $ABC$ in the diagram you can see that $\tan t = \frac{h}{R-r}.$ Solving for $R$ gives

\[R = r + h \tan t.\]

Substitute into the volume expression

\[v= \frac{\pi}{3} \times h \times (R \times R + r \times r+ R \times r).\]

This will give a quadratic in $r$. Solve for $r.$

Penny

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.