# How to calculate a truncated cone?

I have two cylinders. First with diameter=40cm and the second with diameter=48cm. I want to connect them with a truncated cone with hight=15cm. How to calculate the angle of the cone?

EDIT:

The angle I am looking for is angle X in that link -> jean-paul.davalan.pagesperso-orange.fr/calc/cone/index-en.html

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There's no angle $X$ on that page. Do you mean $x$? –  joriki Feb 27 '12 at 11:15
correct. I will edit –  mynkow Feb 1 '13 at 16:57

I have linked to an image below that should help shed some light on how to calculate what you desire. Drawing a simple picture is usually the best way to solve geometry questions. (I would attach the image as part of my answer, but as I am a new user I cannot do that yet).

http://imgur.com/R8eJ4

I am assuming by angle of the cone that you were alluding to the angle that the truncated cone makes with it's base, and that is what I have shown below.

It is clear from the diagram we have a triangle with sides of length 15cm and 4cm, using the knowledge that $tan(x) = \frac{opposite}{adjacent}$ we have $tan(x) = \frac{15}{4} = 3.75$. Which when taking the inverse tangent we find that x = 75.06858 degrees.

Based on the revised question asked the angle you are interested in can be calculated as follows:

First we need to calculate the height of the non-truncated code. This is calculated as the height of the truncated cone multiplied by the ratio of the radius base of the cone and the difference in radius of the base and the top of the truncated cone.

$t = h * \frac{b}{b-a} = 15 * \frac{24}{4} = 90$

Here t is total height of the cone, h is height of the truncated cone, b is radius of the base and a is radius of the top of the truncated cone.

We can now use pythag to find the length of the outside of the full height cone.

$r = \sqrt{t^2 + b^2} = 93.145$

The arc length of the flattened out cone is the circumference of the of base of the cone which is $\pi d$ (or 180d in degrees). So our arc length $l = 180 * 2 * 24 = 8640$

We know that arc length is radius * angle so:

$8640 = l = xr = x 93.145$

rearranging gives $x = 92.9786$

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Not exactly. This one will not create the correct cone. Look angle X in that link -> jean-paul.davalan.pagesperso-orange.fr/calc/cone/index-en.html –  mynkow Feb 26 '12 at 12:40
@mynkow: The angle marked 'x' in the link you gave is used in the construction of a truncated cone from a flat sheet, which you haven't mentioned at all in your question. Please edit your question to include that link and describe what angle you are actually asking for. –  John Bartholomew Feb 26 '12 at 14:56
Hey Q6x, ~ how big rectangle is needed for that flat truncated cone? I will get a big piece of paper first and then cut it :) –  mynkow Mar 3 '12 at 19:28