Three Pyramids in a Cube

Date: 03/04/2002 at 01:23:22
From: Chaitu
Subject: Pyramids and prisms

Hi,

I want to know how 3 pyramids can fit exactly into a rectangular 
prism. My teacher said it was possible, but i can't figure it out.

Date: 03/04/2002 at 09:19:12
From: Doctor Rick
Subject: Re: Pyramids and prisms

Hi, Chaitu.

They won't be symmetrical pyramids: their peaks won't be above the 
middle of the base, but above one corner of the base. Here is a cube:

   _________________________
  |  \                      |  \
  |     \                   |     \
  |        \                |        \
  |           \_____________|___________\
  |           |             |           |
  |           |             |           |
  |           |             |           |
  |           |             |           |
  |           |             |           |
  |           |             |           |
  |           |             |           |
  |           |             |           |
  |___________|_____________|           |
    \         |               \         |
       \      |                  \      |
          \   |                     \   |
             \|________________________\|


Imagine that the lower right square is the front face of the cube, and 
the parallelogram on the left is the left face of the cube. Now I add 
a "body diagonal from one corner to the opposite corner, and two "face 
diagonals" from the same top corner to the opposite corners on the 
back and right faces of the cube:

   _________________________
  |  \                     /|  \
  |     \                // |\    \
  |        \           /    | \      \
  |           \______/___/__|___________\
  |           |    /    /   |  \        |
  |           |  /          |   \       |
  |           |/       /    |           |
  |          /|       /     |    \      |
  |        /  |             |     \     |
  |      /    |      /      |           |
  |    /      |     /       |      \    |
  |  /        |    /        |           |
  |/__________|_____________|       \   |
    \         |               \      \  |
       \      |  /               \      |
          \   | /                   \ \ |
             \|________________________\|


You can see one square pyramid: its base is the bottom face of the 
cube, and its peak is the top back right corner, where the diagonals 
meet. 

Now if you add one more face diagonal, you will have all the edges of 
all three pyramids that fit together to make the cube. Can you see 
where to add the face diagonal, and where the bases of the other two 
pyramids are?

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