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### Frustum of a Cone

```
Date: 12/09/96 at 00:55:37
From: Bill Blackburn
Subject: Cones ?

Hi,

My question is about cones.  I am trying to find the formula for the
following:

If you take a cone of a given size and cut it from the small end to
the large end, then lay it out on a flat surface, what will the inside
and outside sizes of this flattened-out cone be?

Example:
x = diameter of small end of cone   x is 3"
y = diameter of large end of cone   y is 6"
z = length of cone                  z is 2"

Any help or suggestions would be greatly appreciated.  Thank you.

Bill Blackburn
```

```
Date: 12/11/96 at 21:12:03
From: Doctor Rob
Subject: Re: Cones ?

Interesting problem!

Actually, you seem to be dealing with what is called a "frustum" of a
cone.  A cone usually refers to a figure formed as follows: Start
with a plane and a circle lying in it. The circle is called the base
of the cone. Construct the line perpendicular to the plane and
passing through the center of the circle. Choose a point, called the
vertex, on that line but not in the plane. Now consider all the lines
connecting the vertex to points on the starting circle. The distance
from the vertex to the center of the base is called the height of the
cone. The distance from the vertex to the points on the circle is
called the slant-height of the cone.

A frustum of the cone is gotten by cutting the cone with a plane
parallel to the base and lying between the vertex and the base, and
discarding everything on the same side of this cutting plane as the
vertex.

If I understand your question properly, you get a figure in the plane
which looks like this:

______________
/-.    d     .-\
/   `--....--'   \
/       Pi*x       \
s/                    \s
/                      \
/________________________\
`-.         D          .-'
`--...________...--'
Pi*y

The upper arc length is Pi*x, the lower arc length is Pi*y.

I have a small problem with your term "length" here.  I will assume
you mean the height of the frustum, which is the perpendicular
distance between the two bases.  This is not the same as the slant-
height, which would be the length s of the two sides in the above
diagram.

If you extend the frustum to a full cone, then unroll it in the plane,
you would have the same effect as extending the two sides in the above
diagram until they meet and point P.  Call the angle formed by these
two extended lines A.  Then the upper and lower arcs are arcs of
circles centered at P.

If we look at a cross-section of the cone taken through the center
line, we would get a diagram like this:

P
/|\
/ | \
/  |  \
/   |   \
/    |    \
/     |     \
/      |h-z   \ S
/       |       \
/________|________\
/   x/2   |   x/2   \
/          |z        s\
/           |           \
/____________|____________\
y/2         y/2

If we let h be the height of the entire cone, then using similar
triangles, we can get the ratios h/(y/2) = (h-z)/(x/2), which we can
solve for h = y*z/(y-x) and h-z = x*z/(y-x).  The slant-height of the
frustum is then s = Sqrt[z^2 + (y-x)^2/4], which is the length of the
two sides in the first diagram.

If you want to know the length D of the horizontal line in the first
diagram, that will tell you the width of the figure.  It can be
computed as the slant-height of the full cone times 2*sin(A/2).  The
slant-height of the full cone from the second diagram is
S = Sqrt[h^2 + y^2/4], and the angle A is given by Pi*y/S = Pi*x/(S-s).
D = 2*S*sin(Pi*y/(2*S)). Similarly, d = 2*(S-s)*sin(Pi*y/(2*S)).

The height of the first diagram, from the line marked d to the arc
marked Pi*y, would be given by S - (S-s)*cos(Pi*y/(2*S)).

If you don't understand this, or if I have misinterpreted your
clarification.

-Doctor Rob,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry
High School Practical Geometry

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