You have an equilateral triangle you want to wrap with a square piece of paper. If the triangle has unit sides, what is the smallest square piece of paper that can be folded around the triangle such that both the front and back sides of the triangle are completely covered by the paper?

The drawing is very messy on paper, but I will break it down into little parts to that I can draw it here.

First of all, one of the triangle's corners is pointing directly NE.  The other two corners are touching the sides of the square, but we do not yet know how big the square is.

|       _/|
|     _/  /
|   _/   |
| _/     /    I know this is messy, but it's
|/      |     much, much better than nothing
|\      /
| \    |
|  \   /
|   \ |
|    \/    

Let's zoom into the SW corner, and go ahead and fold that corner onto the triangle.

|65/
| /15 
|/____
 \45  |
  \   |   /    These numbers are angle measurements
   \45|15/
    \ | /65
     \|/___

So that I can label some points, let's look at the entire square, ignoring the triangle.

F______D
|      |
E___B  |
 \  |  |
  \ |  |
   \A__C

What I plan to do is fold corner C over the triangle.  Angle BAC must be 50 degrees, and to minimize the square's size, CD must cross B.  AE is not shown in this next diagram.

|
|
E______B
      /|
     / |
    /40|
   /   |    
  C 90 |    /
   \   |   /
    \50|15/
     \ | /
      \|/
       A

So BAC is a 40, 50, 90 triangle.  Similarly, I should be able to fold corner F over the triangle, and everything will be covered, and the three folds intersect at B.

Calculations:
AB = 1 / sqrt(2) = .7071067812
AC = ABsin(40) = .4545194777
Side = AB + AC = 1.161626259

This is not necessarily the best solution

Posted by Tristan on 2005-10-12 16:41:42