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

### Inscribed Cone

```
Date: 3/11/96 at 18:47:33
From: Edward Tang
Subject: Calc problem

Hi -

My Calculus teacher and I are having problems with this question:

"Find the volume of the largest right circular cone that can be
inscribed in a sphere of radius 'r'"

It seems hard to relate 'r' with anything on the cone......
If that connection can be made, then the derivative can be found
and the problem solved......

Thanks!

- Ed
```

```
Date: 6/21/96 at 11:48:14
From: Doctor Jerry
Subject: Re: Calc problem

Start by drawing a triangle inscribed in a circle of radius r.
Draw it as a somewhat slender isoceles triangle, with the base
between the equal sides placed on the bottom.  Locate the center
of the circle.  Draw a line from the center to the lower left
vertex and a line from the center to the base and perpendicular
to it.  You see a small right triangle. Let its lower base be called R.
This is the base radius of the inscribed cone.

The height of the inscribed cone is r plus the vertical side of the
small right triangle.  This side is sqrt(r^2-R^2).

So, the volume of the cone, in terms of r and R, is
V=(pi/3) R^2 (r+sqrt(r^2-R^2)).

This function can be maximized without much trouble.
I get R=2*sqrt(2) r/3.

-Doctor Jerry,  The Math Forum

```
Associated Topics:
High School Calculus

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