# The ultimate conversion of a square into right angle pyramid

This is a follow up of other puzzles. Here a general case of which the other cases are a subset.

Given a square of any size, cut it into four pieces to be reassembled into a right angle pyramid (the apex above one of the base vertices).

One of the cuts must be through the point E that may be positioned any where on the side DC.

## 1 Answer

We can do it as follows

Let the length of the side of the square be $$1$$ and $$|DE| = x$$.
Pick the point $$E'$$ on side $$CB$$ such that $$|CE'| = x$$.
Make the three straight line cuts between $$E$$ and $$E'$$, $$E'$$ and $$A$$, $$A$$ and $$E$$ as shown in the following diagram
Then the fours triangles can be rearranged to form the faces of a right-angled pyramid.
In Cartesian coordinates in $$\mathbb{R}^3$$ we can set the vertices of this pyramid at the points $$(0,0,0)$$, $$(1,0,0)$$, $$(0,x,0)$$ and $$(0,0,1-x)$$.