# Find the height of a cone given that a cylinder of radius 6 and height 6 fits in perfectly.

A cylinder of radius 6cm and height 6cm fits perfectly inside a cone, leaving a constant ring of width x around the base of the cylinder. Find the height of the cone. Here is an image, showing the problem:

I can guess that pythagorus may have to be used, but I am not sure if I need to know some property about ratios of the sides or something!

Many thanks.

• Similar triangles, $6+x$ and $h$ for the big one, $6$ and $h-6$ for the small one.... – Ned Nov 23 '19 at 19:00

$$\frac{6}{x} = \frac{h}{x+6}.$$
$$h=6+\frac{36}{x}.$$