You are to construct a pyramid with a square base that fits into a cube as in the picture above. Design a two-dimensional pattern, that after cutting it out and folding it, you obtain the desired pyramid. Make the base of the pyramid a square with sides 5 centimeters. Determine, without measuring but with the aid of the Pythagorean theorem, the lengths of all the sides of your pattern (except for the 5 cm sides), and show how you found the lengths.
Three of these pyramids can be assembled to form a cube. After building your pyramid, join with two others in the class and take your three pyramids and see how they can be formed into a cube.