Cube in a Cone

Date: 12/17/2002 at 18:17:10
From: Jackie
Subject: Cube inside a cone

I have a cone with a cube inside of it. The radius of the bottom of 
the cone is 1 cm.; the height of the cone is 3 cm. The cube touches 
the cone at the four top corners and hits somewhere toward the middle 
of the cone. The bottom of the cube rests directly in the center of 
the base of the cone but does not touch the circumference of the base. 

I need to know how big each side of the cube is, based on this 
information. I also need to be able to explain how I arrived at the 
answer. 

Date: 12/18/2002 at 01:51:22
From: Doctor Greenie
Subject: Re: Cube inside a cone

Hello, Jackie -

Let the length of the side of the cube be x.

Then the height of the cube is x, so the distance from the top of the 
cube to the top of the cone is (3-x).

The four vertices of the cube that touch the cone lie on a circle 
that is in a plane parallel to the plane containing the base of the 
cone; the diameter of this circle is the diagonal of the face of the 
cube, or x*sqrt(2).

If we now look at the two-dimensional figure we get when viewing the 
cone directly from the side, the entire cone and the portion of the 
cone above the cube appear as similar triangles. From the heights and 
bases of these two similar triangles, we get

   3-x   x*sqrt(2)
   --- = ---------
    3        2

You can solve this proportion to find the value of x.

I hope this helps. Please write back if you have any further questions 
about any of this.

- Doctor Greenie, The Math Forum
  http://mathforum.org/dr.math/