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### Diameter of the Base of a Cone

```
Date: 08/12/98 at 10:58:25
From: ALAN BROWN
Subject: Diameter of the Base of a Cone

I am having difficulty remembering and/or finding the formula that
would allow me to calculate the diameter of the base of a cone of nine
degrees at various lengths. Could you help?

The question would be: how many feet wide is the "footprint" of a cone
of nine degrees at various depths or distances?

Thanks,
Alan Brown
```

```
Date: 08/12/98 at 14:02:07
From: Doctor Mike
Subject: Re: Diameter of the Base of a Cone

Hi Alan,

This is an interesting question, because you can also ask it without
using either the word cone or the word diameter! If you look at the
cone from the front (like an ice-cream cone standing upside-down on a
table, and your eye is about an inch above the plane of the tabletop)
it looks like an isosceles triangle. Isosceles means 2 equal sides.

/\
/18\
/    \
/      \       H feet high
/        \
/          \
/            \
/______________\

I am guessing that 18 degrees is the entire angle at the top. Usually
in discussions about cones we use the "half-angle" which is the angle
between the sloping side and the vertical. If the full angle is 9, the
half-angle would of course be 4.5. Let's split the isosceles triangle
down the middle to get a right triangle.

|\
|9\
|  \
|   \
H |    \
|     \
|      \
|_______\
R

Now we need to introduce some trigonometry into our problems. The
tangent of 9 degrees is Opposite over Adjacent, or in this case R/H.
Because you know H and want to know R, the most useful version of the
formula is:

R = H * tan(9)

Actually, that would correspond to the Radius of the base of the cone,
and what you really want is twice that for the diameter, or:

D =  2 * H * tan(9)

What's that? You don't know the tangent of 9 degrees? Whip out your
calculator or use 0.15838444, which should be close enough. If you need
tan(4.5), that is 0.078701707, but go ahead and buy yourself a simple
scientific calculator. You can probably get one for under \$10.

I hope this helps.

- Doctor Mike, The Math Forum
Check out our web site! http://mathforum.org/dr.math/
```
Associated Topics:
High School Conic Sections/Circles
High School Geometry
High School Higher-Dimensional Geometry
High School Triangles and Other Polygons
High School Trigonometry
Middle School Conic Sections/Circles
Middle School Geometry
Middle School Higher-Dimensional Geometry
Middle School Triangles and Other Polygons

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