School Name
Mathematics
Worksheet 2
Rotational Symmetry in Cubes and Regular Tetrahedra
Name：
Class：
(
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Section 0: Revision on reflectional symmetry in 2D Figures
In each of the following rotational symmetric figures, mark a “” at the centre of rotation and point
out the number of folds of rotational symmetry in each figure:
Square
Equilateral triangle
Rectangle
Hexagon with parallel
edges
____-fold symmetry
____-fold symmetry
____-fold symmetry
____-fold symmetry
Section 1: Rotational Symmetry of Cubes
Part A:
1.
In this section, we will use the model of punched cube and a straw in “Symmetry of 3D
Figures” Package as aids to explore the rotational symmetric properties of a cube. You may
first prepare the model, and pierce the punched cube according to the instructions given in the
questions below. The straw will serve as the axis of rotation.
P
X
Q
Y
Figure 1a
2.
a.
Figure 1b
Figure 1c
As shown in Figure 1a, if you are observing the cube from vertically above the axis XY,
what plane figure will you see? ___________________
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Rotational Symmetry in Cubes and Regular Tetrahedra
b.
Worksheet 2
Observe the rotation of the cube right above the axis XY. Point out what you find.
State the number of fold of rotational symmetry of the cube with respect to XY?
_____________________________________________________________
3.
a.
Refer to Figure 1b, turn the model until the axis PQ is perpendicular to the desktop, then
use only one eye to observe the cube from vertically above the axis PQ (as shown in
Figure 1d) , what plane figure will you see? Does this figure have 4-folded rotational
symmetry? ________________________ (Yes / No).
b.
Keep the axis PQ vertical to the desktop. Observe the cube right above the axis PQ.
Join all holes on the same horizontal plane (such as holes T, U and V) with straight lines.
What will be the rectilinear figures formed by the lines?
______________ triangle;
_________________
T
U
V
Figure 1d
c.
How many folds of rotational symmetry do these figures have? Deduce the number of
folds of rotational symmetry of the cube with respect to PQ through your observation to
the model. Briefly describe the strategy that you use to find the number of folds of
rotational symmetry.
4.
There is another type of axis of rotational symmetry for cubes.
above methods. Sketch the axis in Figure 1c.
Try to explore it using the
5.
Sketch one axis of rotational symmetry on each cube provided on next page and give the total
number of axes of rotational symmetry.
____________________________________________________________________________
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Rotational Symmetry in Cubes and Regular Tetrahedra
H
H
G
E
Worksheet 2
E
F
C
H
B
D
A
D
G
E
F
C
C
D
D
H
G
F
C
C
B
H
B
D
A
G
E
F
C
F
C
B
A
A
H
G
E
C
B
D
A
H
G
F
G
E
F
C
H
G
E
F
A
A
A
H
B
B
B
D
A
H
F
C
E
D
G
E
F
B
A
G
D
F
C
H
H
E
E
B
A
D
G
C
B
E
H
F
C
D
A
G
E
F
B
D
A
H
G
E
F
C
B
D
G
E
F
C
H
G
B
D
A
B
D
A
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Rotational Symmetry in Cubes and Regular Tetrahedra
Worksheet 2
Part B: Conclusion
1. The number of folds of rotational symmetry is the same / different with respect to different
types of axes of rotational symmetry.
2. There are totally ____ axes of rotational symmetry for a cube. The axis passes through two
points of the surface of a cube, which can be the cube’s ___＿＿＿＿, ＿＿＿＿＿＿＿__ or
＿＿＿＿＿＿＿＿.
3. By putting “✓” into appropriate boxes and filling in all the information required, complete the
following table:
Axis of rotational symmetry
Point 1*
Type
Vertex
Point 2*
Centre
Centre
of face
of edge
Vertex
Number of folds
Centre
Centre
of rotational
of face
of edge
symmetry
Quantity
Type 1
Type 2
Type 3
Total
*Tick the appropriate boxes
Table 1
Section 2: Rotational Symmetry of Regular Tetrahedra
Part 1:
1. Using similar methods in Section 1, you can try to explore the rotational symmetric properties
of regular tetrahedra. You may use the punched regular tetrahedron models and straws in the
“Symmetry of 3D Figures” Package as aids.
2. In the figures provided, draw all the axes of rotational symmetry for regular tetrahedra. In
each figure, you should also write down the number of folds of rotational symmetry and mark
down  at where the axis passes through the surface of the tetrahedron (as illustrated in the first
two figures):
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Rotational Symmetry in Cubes and Regular Tetrahedra
Worksheet 2
X
A
A
X
D
D
B
B
Y
C
C
_____-fold of symmetry _____-fold of symmetry
_____-fold of symmetry
Y
_____-fold of symmetry
A
A
D
B
A
D
B
A
D
D
B
B
C
C
C
_____-fold of symmetry
_____-fold of symmetry
_____-fold of symmetry
C
_____-fold of symmetry
Part 2: Conclusion
1.
There are totally ____ axes of rotational symmetry for a regular tetrahedron. The axis passes
through two points of the surface of a tetrahedron, which can be the tetrahedron’s _＿＿＿＿,
＿＿＿＿
＿＿＿ or ＿＿＿＿＿＿＿＿.
2.
Using the positions where the rotational symmetric axes pass through the surface of the
tetrahedron, one can categorise the axes into a few types. Use the results in Part 1 to
complete the following table:
Axis of rotational symmetry
Point 1*
Type
Centre
Point 2*
Centre
Vertex
Number of folds
Centre
Centre
of rotational
of face
of edge
symmetry
Vertex
of face
of edge
Quantity
Type ___
Type ___
Type ___
Total
*Tick the appropriate boxes
Table 2
~End~
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