# Unfold a right angle pyramid into a square

This puzzle refers to a feature of right angle pyramid:

The relation between the areas of the three perpendicular faces and the diagonal surface area is given as - $$S^2_x+S^2_y+S^2_z = S^2_d$$

Visit the link for details: De Gua's theorem

1. The challenge is to unfold the 3D pyramid surfaces into a 2D shape and than cut it into two pieces to be reassembled into a square. There is one specific case where unfolding will create a square with no need to cut the shape.

2. It is not possible for all pyramids (I think) - what is the condition with regard to the surfaces for this to be solved?

• Comments are not intended for discussion - and particularly not for discussion of an entirely different question. If you want to discuss a question's closure, try asking about it on our Puzzling Meta. – Rubio Jul 17 '19 at 0:21
• Can you define "right angle pyramid" for me, please? – Dr Xorile Sep 5 '19 at 18:23
• Very similar to this question: puzzling.stackexchange.com/questions/86082/… – Dr Xorile Sep 5 '19 at 18:26
• Similar but not the same. – Moti Sep 6 '19 at 19:14

## 1 Answer

Here's a partial answer, which is for the case where unfolding will create a square with no need to cut the shape:

Proof: ABCD is a square, so we need to show that it can be folded to a right pyramid. Bisect DC at E. Bisect BC at F. Connect AE, AF, and EF. Then fold △AED, △ABF, and △EFC up (or down), so that C, D, and B meet. This can be done because EC=ED, CF=BF, and DA=BA. And clearly angles, D, C, and B are right angles.

Depending on definitions, there is a class of pyramids for which this can be done using the disection technique:

The above shows that any rectangle can be folded into a right pyramid using a similar proof (except the base is a quadrilateral rather than a triangle). Then any rectangle that can be cut and reformed into a square can be done. For example, a 9x16 rectangle can be cut into steps to form a 12x12.

• Nice for the case of not cutting. The second solution is nice but not meeting the requirement of right angle pyramid - the provided equation refers to it - three sides are right angle triangles. – Moti Sep 6 '19 at 19:16
• The question also asks how to do it by cutting into two pieces - hint: another puzzle I posted - many such pyramids may be created. – Moti Sep 6 '19 at 19:18