Chapter 8: The Square-based Pyramid and the Cube
Activity 8.1: At home, construct the following square-based pyramid out of posterboard using the following pattern:
Fig 8.1
When drawing the pattern you should also draw tabs to be able to glue it together.
In this case 4 tabs are enough (you can draw them alternately, one on each triangle).
Once the pyramid is assembled it should look like this (among other views):
Fig 8.2
Bring your pyramid to the next class.
Activity 8.2: In class, you will work in pairs with two pyramids (one from each student
in the pair) and try to form new polyhedrons that can be characterized by counting the
number of their faces, vertices and edges.
Shown here are all the possible convex polyhedrons that could be obtained:
1) A Pentahedron (rectangular base pyramid)
Fig 8.3
We join two pyramids
Fig 8.4
We eliminate interior vertices
and edges
Fig 8.5
Here we have a representation of the polyhedron obtained by joining two pyramids in
the manner illustrated in Fig 8.3.
Note: You can make it with Cabri 3D by reflecting the pyramid over the common face.
2) Another pentahedron (here, we have again three types of representations)
Fig 8.6
Fig 8.7
Fig 8.8
3) A hexahedron (a bi-pyramid)
Fig 8.9
Fig 8.10
Fig 8.11
(This representation was obtained using Cabri 3D by reflecting a pyramid about its square base)
4) A heptahedron
Fig 8.12
Fig 8.13
Fig 8.14
5) An octahedron
Fig 8.15
Fig 8.16
(This representation can be obtained by
the point reflection of the pyramid
through the center of the square)
Fig 8.17
A non convex-heptahedron (two thirds of a cube – a third pyramid would complete the
cube – see the following combination)
Fig 8.18
Homework 8.1: Find out, in all cases, whether these bodies satisfy or not Euler’s
Formula.
Homework 8.2: Instead of the square-based pyramid you constructed in Activity 8.1,
construct an equilateral square-based pyramid with equilateral triangles as the four
slant faces. Use an edge-length the same as the regular tetrahedron that you created for
Chapter 2 (10 cm). Now join this equilateral square-based pyramid to your regular
tetrahedron. How many faces will your square pyramid + tetrahedron have? Read the
TIME article concerning an error on the SAT test with respect to this question and write
a substantial comment (CD file: TIME_SAT_error.pdf).
Activity 8.3: You should now form groups of three students and discover new bodies
using three pyramids.
If you join together your three pyramids as in the following picture, what solid will be
formed?
Fig 8.19
Fig 8.20
A cube with edge 7 cm?
While figure 8.20 looks like a cube, you need to convince each other that the three
pyramids altogether do, in fact, form a cube and as a consequence the volume of each
pyramid is a third of the cube’s volume.
The volume of a cube of edge “a” is:
Vcube = a.a.a =
=
a3
⇒
. .
3
Vpyramid
a3
=
3
!
and in this way, without having done a formal proof, but having done a meaningful
construction, you discover that the following formula works for this particular pyramid:
Vpyramid =
AreaBase . Height pyramid
3
Homework 8.3
Find a proof or demonstration that the above formula holds for the volume of any
pyramid. You will need this formula to find the volumes of the four none-cube Platonic
Solids in the next chapter.
Homework 8.4
The set of regular polygons is an infinite set; however the set of the regular polyhedrons
is finite and has only 5 elements. It is hard to believe, but we can count the regular
polyhedrons with the fingers of just one hand!
Why are there only 5 regular convex polyhedron? Develop a rationale explanation for
this fact (i.e. a proof).
Hint for a proof about the existence of only 5 regular polyhedra:
Any regular polyhedron, be definition, has faces that are regular polygons. Let’s now
see which polyhedrons we can build with the first type of regular polygon, which is an
equilateral triangle:
Fig 8.21
If three triangles meet at a vertex, the sum of their angles is:
60 x 3 = 180° and 180° < 360°
Thus, if we fold the triangles at that vertex to form a solid, we have a polyhedric angle
that corresponds to the regular tetrahedron:
Fig 8.22
If 4 triangles meet at a vertex
Fig 8.23
60 x 4 = 240° < 360°
we have the polyhedric angle of a regular octahedron from which we can see two
different representations in the following figures:
Fig 8.24
Fig 8.25
We can continue trying with 5 equilateral triangles around a vertex and then 6
equilateral triangles, checking that the sum of the angles of the figures around a vertex
is smaller than 3600 (because, if the angle is equal to 3600, we would have a flat surface
or a tessellation of the plane.
Once we have exhausted the possibilities with equilateral triangles, we have to try the
same procedure with squares, regular pentagons, regular hexagons and so on.