Fascination with Pyramids again...

Do you recall the post about pyramids from last April? This is a continuation of that investigation and is a problem that has appeared on standardized tests. There are several approaches and the learning objectives for geometry students are many:


(1) Develop spatial reasoning
(2) Review terminology of space figures, pyramids in particular
(3) Make connections to the real-world problem of finding the height of an Egyptian pyramid
(4) Apply the Pythagorean Theorem or special right triangles
(5) Justify (prove) one's methods

STUDENT/READER PROBLEM
The figure attempts to depict a special regular square pyramid.
The 4 lateral edges and the 4 base edges all have the same length x.
Show that the height PT of the pyramid has length x(√2/2).

Notes:
(a) It may be instructive to encourage students to approach this by more than one method. One could ask students to find 30-60-90 as well as 45-45-90 triangles in the pyramid (lines may need to be constructed of course).
(b) Many students may assume ΔPTS is 45-45-90. Challenging them to prove it is an important objective here (there's more than one way).
(c) One could begin with a specific value of x, such as x = 10 (see the original pyramid post).
(d) I strongly urge you to have students research the Great Pyramid of Giza. Is it approximately a regular square pyramid? Are its faces equilateral triangles as in this post? There are many classic math problems associated with this Wonder of the World and it's not all about geometry!

Pyramid Power: An Investigation that Develops Spatial Reasoning with Pyramids, Nets, Constructions,...

This investigation focuses on regular square pyramids, i.e., those with a square base and whose vertex is directly above the center of the base (informally stated).
The questions below are designed to further students 3-D visualization by constructing and 'deconstructing' several of these pyramids. Younger students in upper elementary or middle school should have had several experiences building and manipulating these kinds of solids, long before quantitative considerations of lengths of segments, angles, areas or volumes. Middle schoolers and high school students can always benefit from a hands-on approach to review the basic ideas, and geometry software like Geometer's Sketchpad is also very helpful to explore lengths, angles, surface areas, etc.

1. Draw a regular square pyramid, the kind you would see in Egypt. Give it a 3-D perspective. Each base edge should be 10 units for the rest of this investigation.
2. Draw a net for your pyramid.
3. Based on these drawings, answer the following:
(a) The triangular faces are always __________ triangles.
(b) If the lateral edges (segments from the vertex of the pyramid to a base vertex), are also 10 units, then each triangular face is a(n) _____________ triangle.
(c) Explain why the lateral edges cannot each be 5 units.
(d) Many students would guess that the minimum lateral edge is 10, but in fact it could be less. Finish this statement: The lateral edges must be greater than x. The greatest possible value of x is ________. (Mathematicians would call this greatest lower bound).
(e) If the lateral edges are each 10 units (all faces are equilateral), determine the height of the pyramid.
(f) If the height of the pyramid is 5 (half the base edge), it is easy to show, using a formula, that the volume of this pyramid is one-sixth the volume of the smallest cube containing the pyramid (the base of the cube coincides with the base of the pyramid). However, your task is to explain this visually without any formulas! You could 'build' a few of these pyramids and show that six of them will fit in the cube but is that necessary?
To be continued...