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### Construction of Cone Surface

```Date: 10/12/2002 at 11:00:19
From: Albert
Subject: Construction of cone surface

What is the relation between the angle of a cone at the vertex and the
flat angle (theta) of the developed surface on a plane?

Example: I want to build a cone with a 30-degree aperture angle at the
vertex, height h, and base radius r. What is the angle at the vertex
of the circle sector when I develop that surface on the floor prior to
cutting the material?

Is there a general formula or function theta = f(alpha)?
Alpha = angle at the vertex.

Thank you.
```

```
Date: 10/12/2002 at 23:22:10
From: Doctor Peterson
Subject: Re: Construction of cone surface

Hi, Albert.

This is a common question; I searched our site for "cone surface
sector" and found this, among others, which gives a picture:

Lateral Surface of a Cone
http://mathforum.org/library/drmath/view/55082.html

Let's copy that picture and add a side view of your cone:

*********
******         ******
***                   /
**                     /
**                      /
*                       /
*                       /s
*                       /
*                       /
*                      /
*                      /
*                     +---------------------*
*                theta                      *
*                                         *
*                                         *
*                                       *
*                                     *
*                                   *
**                               **
**                           **
***                     *** 2 pi r
******         ******
*********

+
/|\
/ | \
/alpha\
/   |   \
/    |    \s
/     |h    \
/      |      \
/       |       \
/        |        \
+---------+---------+
r         r

The angle theta can be found by taking the ratio of the arc length,
2 pi r, to the circumference of the whole circle, 2 pi s, and
multiplying 360 degrees by that ratio. This is the angle r/s times
360 degrees. Since your angle alpha satisfies

sin(alpha/2) = r/s

you just have to multiply 360 degrees by this sine.

If you have any further questions, feel free to write back.

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

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