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?

  • $\begingroup$ 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. $\endgroup$
    – Rubio
    Jul 17, 2019 at 0:21
  • $\begingroup$ Can you define "right angle pyramid" for me, please? $\endgroup$
    – Dr Xorile
    Sep 5, 2019 at 18:23
  • $\begingroup$ Very similar to this question: puzzling.stackexchange.com/questions/86082/… $\endgroup$
    – Dr Xorile
    Sep 5, 2019 at 18:26
  • $\begingroup$ Similar but not the same. $\endgroup$
    – Moti
    Sep 6, 2019 at 19:14

1 Answer 1


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

unfold a pyramid
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:

unfold a pyramid 2
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.

  • $\begingroup$ 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. $\endgroup$
    – Moti
    Sep 6, 2019 at 19:16
  • $\begingroup$ The question also asks how to do it by cutting into two pieces - hint: another puzzle I posted - many such pyramids may be created. $\endgroup$
    – Moti
    Sep 6, 2019 at 19:18

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.