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Diameter of Cone Frustum

```
Date: 05/24/2000 at 05:39:25
From: Vin
Subject: Diameter of cone

Dear Dr. Math,

I need to solve for the diameter of a cone when it is opened up flat.
The inner and outer diameters, and the height are given. The top part
of the cone is cut of by the inner diameter.

Kindly help me.
```

```
Date: 05/24/2000 at 09:05:42
From: Doctor Peterson
Subject: Re: Diameter of cone

Hi, Vin.

It's not quite clear whether the inner and outer diameters you are
given refer to the bases of the frustum of a cone, or to the flattened
shape; I'll assume you mean you are given D, d, and h in this side
view of a frustum:

d
+----+----+
/     |     \
/      |      \
/       |       \
/        |h       \
/         |         \
/          |          \
/           |           \
+------------+------------+
D

You can find the total height of the uncut cone using similar
triangles:

+ -----------------
/|\              |
/ | \l            |
/  |  \            |
/   |  r\           |
+----+----+          |
L/     |     \         H
/      |      \        |
/       |       \       |
/        |h       \      |
/         |         \     |
/          |          \    |
/           |           \   |
+------------+------------+ ----
R

Then the radii of the circles you get when you flatten the cone are
the slant heights L and l of the cone, which you can find using the
Pythagorean theorem.

Write back if you need more help.

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

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