Associated Topics || Dr. Math Home || Search Dr. Math

### Cone Volume

```
Date: 04/19/99 at 16:59:27
From: SHAMSI
Subject: Volume of a cone

A right circular cone is to be circumscribed about a sphere of radius
R cm. Find the ratio of the altitude to the base radius of the cone of
largest possible volume.

I started this question but got confused because there are no actual
numbers to work with. Your help would be greatly appreciated.

Thanks,
Shamsi
```

```
Date: 04/19/99 at 17:31:27
From: Doctor Rob
Subject: Re: Volume of a cone

Thanks for writing to Ask Dr. Math!

Let the radius of the sphere be r, the radius of the base of the cone
R, and the altitude of the cone h.  Draw this cross-sectional picture:

C
o
/|\
/ | \
/  |  \
/   |   \
/    |    \
/     |     \
/      |      \
/       |       \
/   _,,--+--.._   \
/ ,-'     |h-r  `-. \
/,'        |        `.\
,'          |          `.
/            |            \ D
/           h |         _,-'\
,              |     _,-'     .
/|              | _,-' r       |\
/ +             O+'             + \
/  |              |              |  \
/   .              |              ,   \
/     \             |             /     \
/       \            |r           /       \
/         `.          |          ,'         \
/            `.        |        ,'            \
/               `-._    |    _,-'               \
o--------------------``--+--''--------------------o
A           R             B

Using similar right triangles ABC and DOC, you get the equation

h/R = CD/r
CD = h*r/R

Using the Pythagorean Theorem on DOC, you get

(h*r/R)^2 + r^2 = (h-r)^2

You are interested in the value of x = h/R, so elminate R from the
equation by substituting R = h/x:

(r*x)^2 + r^2 = h^2 - 2*h*r + r^2
r^2*x^2 = h^2 - 2*h*r

Now you can solve this equation for h, and substitute that into the
formula

V = Pi*h*r^2/3

for the volume. Then differentiate the result with respect to x,
remembering that r is a constant. Set the derivative equal to zero,
and solve for x.

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

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search