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### Building a Cone

```
Date: 01/28/2002 at 02:43:43
From: Surendra Kumar Chordia
Subject: Building a larger radius cone

My question is about cones.  I am trying to draw a larger radius size
cone (frustum). The details are as follows:

Example:

x = diameter of small end of cone   x is 6.5 meter
y = diameter of large end of cone   y is 7.0 meter
z = height of cone                  z is 5.0 meter

For the above cones the the inner and outer radius comes out to:

R =  larger radius of developed cone   R is 70.087 meter
r =  smaller radius of developed cone  r is 65.081 meter

the above radii are difficult to draw to 1:1 scale, so please gives
some alternative.

Any help or suggestions would be greatly appreciated.  Thank you.
S.K.Chordia
```

```
Date: 01/28/2002 at 12:58:34
From: Doctor Peterson
Subject: Re: Building a larger radius cone

Hi, Surendra.

It appears that you are forming a frustum by cutting out a portion of
an annulus and rolling it into the lateral surface of the frustum. You
have not told me how you found the radii; I presume you have a formula
such as those explained here:

http://mathforum.org/dr.math/problems/loraine.02.17.01.html

Flattening the Frustum of a Cone
http://mathforum.org/dr.math/problems/scholl.10.15.00.html

I think what you are saying is that you want a convenient way to
draw an arc with a 70 meter radius without having to find a center
70 meters away and use a 70-meter string. The simplest solution I can
think of is to plot a set of points on the arc, through which you can
draw a smooth curve. One way to do this uses the properties of a
segment of a circle:

Circle Radius from Chord Length and Depth
http://mathforum.org/dr.math/problems/duranleau.6.16.99.html

The right triangle in the picture on that page shows that

r-h
--- = cos(theta/2)
r

Assuming you know the angle theta of your arcs (as given in the
earlier references), you can solve this for h:

h = r(1 - cos(theta/2))

The same formula is found here:

Segments of Circles - Dr. Math FAQ
http://mathforum.org/dr.math/faq/faq.circle.segment.html#13

So draw the chord of the appropriate length, find its center, and
measure h units perpendicular to the chord to locate the midpoint of
the arc. Now repeat the process using the two new chords you've found,
repeatedly bisecting the arc until you have points close enough to
form an accurate arc.

There are probably some quick methods you can develop for doing these
calculations more efficiently; I seem to recall a compass and
straightedge construction that can accomplish this, but the
calculations are probably good enough, and more accurate in practice.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Higher-Dimensional Geometry
High School Higher-Dimensional Geometry

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