Question Bank
QB1:
In the lesson some nets of shapes were available to print out. Print them, cut them out
and fold them to make the shapes. What do all of these 3D shapes have in common?
QB1A:
They are all 3D; they all consist of vertices, edges, and faces (an edge is where two faces
meet, and a vertex is a corner (where three or more faces meet)); and they all have only
flat surfaces. These shapes are examples of polyhedra. A polyhedron (pl. polyhedra) is
a special kind of 3D object. It has faces, vertices and edges. All of its faces are flat
surfaces. Pyramids1 and prisms2 are two of many examples of polyhedra.
QB2:
From the prisms and pyramids that you made, can you see any relationship between the
names of the objects and what they look like?
QB2A:
Prisms and pyramids is that they have what’s called a base. The base is the face with the
property that if you cut the solid anywhere parallel to the surface of the base, the same
shape will appear at the cut.
Example:
Identify the base of the following polyhedra. Why is this the base?
a.)
b.)
Solution:
a.) The base is the square. This is because the square is the only face with the
property that if you cut the pyramid anywhere parallel to the surface of the
square, the same shape will appear at the cut (even though it may be smaller).
b.) The base is the pentagon. Again, this is the only face with the property that
cutting the solid anywhere parallel to it will produce a cut of the same shape.
1
2
Link to definition from grade 2, 3, or 4.
Link to definition from grade 2, 3, or 4.
The relationship between the name of the object and what it looks like, is that the first
word of the name gives the shape of the base, and the second word gives the type of solid
it is. For example, for a pentagonal prism the base would be a pentagon and the solid
would be a prism.
QB3:
Complete the following tables by looking at the objects that you made, and then
predicting what may happen with more complicated polyhedra .
PRISM
Number of Sides in Base
Vertices
Faces
Edges
Number of Sides in Base
Vertices
Faces
Edges
Number of Sides in Base
3
4
5
6
8
Vertices
6
8
10
12
16
Faces
5
6
7
8
10
Edges
9
12
15
18
24
Number of Sides in Base
3
4
5
6
8
Vertices
4
5
6
7
9
Faces
4
5
6
7
9
Edges
6
8
10
12
16
Triangular
Rectangular
Pentagonal
Hexagonal
Octagonal
PYRAMID
Triangular
Rectangular
Pentagonal
Hexagonal
Octagonal
Do you notice any patterns?
QB3A:
PRISM
Triangular
Rectangular
Pentagonal
Hexagonal
Octagonal
PYRAMID
Triangular
Rectangular
Pentagonal
Hexagonal
Octagonal
For prisms:
The number of vertices is the number of vertices in the base times two. This is because a
prism is just the base drawn twice, with the vertices connected. The number of faces is
the number of sides in the base plus two (one face for each edge plus the front and back
faces). The number of edges is the number of vertices plus the number of faces, minus
two. In mathematical language, these relationships would be written as follows:
1. # of Vertices = # of vertices in the base x 2
2. # of Faces = # of sides in the base + 2
3. # of Edges = # of vertices + # of faces - 2
For pyramids:
The number of vertices is the number of sides in the base plus one. This is because a
pyramid is just the base with each vertex connected to a point not on the base. The
number of faces is the same as the number of vertices (one face for each of the sides in
the base plus the base itself). Again, the number of edges is the number of vertices plus
the number of faces, minus two. In 1751, a great mathematician by the name of
Leonhard Euler (1707-1783) proved that this is true for all convex polyhedra3. It is now
known as “Euler’s Formula”, and is written as F + V = E + 2 (Faces + Vertices = Edges +
2). This can be rewritten as V + F – 2 = E, which is what we noticed in our tables above.
In mathematical language, these relationships would be written as follows:
1. # of Vertices = # of sides in the base + 1
2. # of Faces = # of vertices
3. # of Edges = # of vertices + # of faces - 2
QB4:
On grid paper, draw two of the polyhedra that you made.
QB4A:
We can draw these polyhedra by following the steps below.
For prisms:
1. Draw a base.
2. Draw the other base.
3. Connect the vertices.
4. Dot the hidden edges, or
5. Leave out the hidden edges.
1.
2.
3.
4.
5.
For pyramids:
3
A convex polyhedron is a polyhedron for which a line connecting any two points (that don’t lie on the
same face) on the surface always lies in the interior of the polyhedron.
1.
2.
3.
4.
5.
1.
Draw the base.
Draw a dot for the point.
Connect the point to the vertices.
Dot the hidden edges, or
Leave out the hidden edges.
2.
3.
4.
QB5: Complete the following drawings of prisms.
QB5A:
We can use the steps we learned for drawing prisms to help us here.
1. Draw a base.
2. Draw the other base.
3. Connect the vertices.
4. Dot the hidden edges, or
5. Leave out the hidden edges.
5.
QB6: Complete the following drawings of pyramids.
QB6A:
We can use the steps we learned for drawing pyramids to help us here.
1.
2.
3.
4.
5.
Draw the base.
Draw a dot for the point.
Connect the point to the vertices.
Dot the hidden edges, or
Leave out the hidden edges.
QB7:
What’s wrong with the following models of polyhedra?
QB7A:
None of these objects could actually be constructed. They are known as ‘impossible
objects’. Look closely at each of them and you will see why.
QB8:
For a 10-sided prism how many edges, faces, and vertices do you think it has? Try doing
this without actually counting. What about 12-sides? What about a 10-sided pyramid?
QB8A:
A 10-sided prism (also called a decagonal prism) has 20 vertices (twice the number of
vertices in the base), 12 faces (the number of sides in the base plus two), and 30 edges
(vertices + faces – 2). Similarly a 12-sided prism (dodecagon) has 24 vertices, 14 faces,
and 36 edges. For a 10-sided pyramid there are 11 vertices (number of sides in the base
plus one), 11 faces (same as the number of vertices), and 20 edges (vertices + faces – 2).
QB9:
Complete the following drawings.
QB9A:
QB10:
The top, front, and side views of some objects are given below. Draw the objects.
1.
Top
Front
Side
2.
Top
Front
Side
Front
Side
3.
Top
QB10A:
1.
2.
3.
QB11:
The faces of three polyhedra are given below. Name the polyhedra that the faces belong
to.
a.)
b.)
c.)
QB11A:
a.) pentagonal prism
QB12:
b.) triangular pyramid
c.) rectangular pyramid
Can you rearrange the six matchsticks below to form four triangles?
QB12A:
Make them into a triangular pyramid as shown below.
QB13:
What’s wrong with the following picture?
QB13A:
The object shown could not actually be built, and you can see why if you look at it
closely. The picture is of the famous “impossible staircase”. It appears as though one
could walk up or down forever.
Below is another impossible staircase.
QB14:
Is a cylinder a polyhedron? Why or why not?
QB14A:
A cylinder is not a polyhedron because not all of its faces are flat surfaces.
QB15:
State Euler’s Formula. How is this formula useful?
QB15A:
In any convex polyhedron, # of Vertices + # of Faces = # of Edges + 2. This is useful
because we will never have to actually count edges. Once we know how many vertices
and faces there are, we can just use the formula.
QB16:
Below are the front faces of some prisms. Complete the drawings (they don’t have to be
any particular size).
QB16A:
QB17:
Give some real life examples of polyhedra.
QB17A:
The Egyptian pyramids (example of rectangular pyramids), a book (example of a
rectangular prism), a box of Toblerone chocolates (example of a triangular prism); there
are many examples of polyhedra in the world.