F.3 Mathematics Supplementary Notes
Chapter 7 More about 3-D Figures
Important Terms
Reflection symmetry
Plane of reflection
Rotational symmetry
Axis of symmetry
Net
Three Dimension
Euler’s Rule
Edge
Vertex
Projection
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反射對稱
對稱平面
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五邊形
六邊形
Pentagon
Hexagon
旋轉對稱
對稱軸
摺紙圖樣
正多面體
柏拉圖立體
索瑪立方塊
Regular Polyhedron
Platonic Solids
Soma Cube
三維空間
尤拉公式 / 歐拉公式
邊
頂點
投影
正四面體
正六面體
正八面體
Tetrahedron
Hexahedron
Octahedron
正十二面體
正二十面體
Dodecahedron
Icosahedron
Revision
1. Two Dimension symmetry
Definition
Examples
Reflectional symmetry
A shape is symmetrical if both sides are
the same when a mirror line is drawn.
(Fig. a)
(Fig.b)
(Fig.c)
The dotted lines cut the figures in two half, each half is a mirror image of the
other side, the dotted line is called the axis/axes of symmetry. Some figures
have more than one axes. (Fig. b and c)
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Rotational symmetry
A figure has rotation symmetry if you can
rotate (or turn) a figure around a center
point by fewer than 360° and the figure
appears unchanged, then the figure has
rotational symmetry.
The point around which you rotate is
called the center of rotation, and the
smallest angle you need to turn is called
the angle of rotation. If a shape will only
fit itself in one way it has no rotational
symmetry.
This figure has rotation symmetry of 72°, and the center of rotation is the center
of the figure
2.Three Dimension symmetry
Definition
Reflectional symmetry is also called
plane symmetry in 3D. In the examples,
the shaded plane work as a mirror to cut
the figures in equal halves, the shaded
plane is called “plane of reflection”.
Examples
Error!
A Cube has 9 planes of reflection
Error!
A Tetrahedron has 6 planes of reflection
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Rotational Symmetry
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Error!
When an object fits exactly with the original
figure n times during one revolution about
a fixed line, the object is said to have
rotational symmetry of order n or n-fold
rotational symmetry. The fixed line is
called the axis of rotation.
D
HG
A
F
C
E
B
A tetrahedron has 4 rotational symmetric axies. The dotted lines represents
axies of symmetry for the figure.
3. Net
Error!
3D figures could be form by net (2D 
3D). A 3D figure can be folded from
different nets.
2.
The technique to form a 3D figure requires some lines to joint together. Normally
these lines that joint together are of equal length.
(a)
(b)
Error!
Lines to joint ;
BA &HA，BC &DC，
Points that sticks together
B
DE &FE，FG &HG
B、D、F &H
H
C
A
H
F
A
D
E
G
F
B D
G
C
E
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4. 2D representations of 3D objects
Error!
We can draw the front, back, top, left
and right views of a 3D object to
describe the shape of the 3D object.
Usually it needs at least three 2D
representations, including top, side
and front views in order to identify or
draw a 3D object.
Front view
1
2
5. Relationship between a straight line and a plane
The angle  that forms between AB and the projection of
AB (i.e. AC) is called the angle between AB and the plane .
Top view
Right view
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6. Relationship between two planes
 is an inclined plane and  is a horizontal plane.
The line PQ, which is perpendicular to the line of
intersection AB, is called the line of greatest slope in plane .
AB is the line of intersection of two intersecting planes  and .
Both lines PQ and RQ are perpendicular to AB at Q.
The acute angle  formed between PQ and RQ is
the angle between two planes  and .
7. Regular Polyhedra
If the faces of a polyhedron are identical regular polygons and that the same number of faces meet at each vertex,
then the polyhedron is a regular polyhedron.
8. Euler’s Formula
All polyhedra satisfy the Euler’s
formula: F + V  E = 2
F is the number of faces
V is the number of vertexes
E is the number of edges
Error!
(a)
Face (F)
Vertex (V)
Edge (E)
F+V–E
6
8
12
2
6
5
9
2
9
9
16
2
(b)
(c)
F + V E = 2
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9. Platonic Solids
The Platonic solids are convex polyhedra with regular polygon faces. Faces and vertices are identical. There are only
five of them, shown below.
Exercise
Level One
1.The figure shows a prism with a trapezium base. How many plane(s) of reflection is/are there? Show
all the plane(s) of reflection by drawing diagram(s).
2.The figure shows a right prism with an isosceles right -angled triangular base. How man y plane(s) of
reflection is/are there? Show all the plane(s) of reflection by drawing diagram(s).
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3.Draw the front, top and side views of the solid.
Top view
Side view
Front view
________________________________________________________ __
TOP VIEW
FRONT VIEW
SIDE VIEW
4.Draw the front, top and side views of the solid.
Top view
Side view
Front view
______________________________________________________
TOP VIEW
5.
FRONT VIEW
SIDE VIEW
A
B
In the figure, ABCDEFGH is a cuboid.
(a) Find the projection of A on the plane EFGH.
(b) Find the projection of A on the plane CBGH.
(c) Find the projection of A on the plane CDEH.
C
D
G
F
(d) Find the projection of AD on the plane CBGH
(e) Find the projection of AD on the plane EFGH
(f)
6.
Find the projection of AB on the plane EFGH
With the same diagram in question 5.
(a) Name the angle between the line AH and the plane EFGH.
(b) Find the projection of A on the plane CBGH.
(c) Find the projection of A on the plane CDEH.
(d) Find the projection of A on the plane BCGH.
E
H
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7.
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The figure shows a right triangular prism.
(a) Find the projection of the line segment DC on the plane ABFE.
D
C
E
(b) Find the projection of the line segment AC on the plane ABFE.
F
(c) Find the projection of the line segment DE on the plane ABEF. A
B
(d) Find the projection of the line segment AF on the plane CDEF.
8. With the same diagram in question 7.
(a) Name the angle between the line AC and the plane EFGH.
(b) Name the angle between the line AC and the plane BCF.
(c) Find the projection of the line segment AB on the plane CDEH.
(d) Find the projection of the line segment AD on the plane ACF.
Multiple Choice Questions
Ans:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
1. Which of the following is a right pyramid?
I.
II.
III.
IV.
A. II only
B. III only
C. II, III and IV only
D. I, II, III and IV
2. Which of the following are correct?
I.
The base of a right triangular pyramid is a triangle.
II.
The lateral face of a right rectangular pyramid is a triangle.
III.
The lateral face of a right regular pentagonal pyramid is a regular pentagon.
A.
I only
B.
II only
C.
I and II only
D.
I, II and III
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
3. How many planes of reflection are there in a right regular 21 -gonal pyramid?
A. 21
B. 22
C. 23
D. 24
4. If the following net is folded up to form a regular polyhedron, what kind of solid will you get?
A. Regular tetrahedron
B. Regular hexahedron
C. Regular octahedron
D. Regular dodecahedron
5. Which of the following is not a net of a regular hex ahedron?
A.
B.
C.
D.
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6. Letters A, B, C, D and E are printed on each face of a cube, and one of t he letters appears twice.
The figures below show three different ways to place the cube, which of the following is a net of
the cube?
A
E
A
D
C
A
A
B
Figure I
C
B
A
D
C
D
A.
B.
B
A
D
B
C
Figure II
Figure III
D
E
C.
E
C
B
C
A
D
D.
C
D
A
B
E
D
7. Which of the following nets can be folded up to form the regular octahedron be low?
A.
B.
2
1
4
2
3
4
3
4
1
4
2
1
3
3
1
2
C.
D.
2
1
4
3
2
3
2
4
1
4
2
1
4
3
3
1
1
2
3
4
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8. Which of the following solids has/have the same front view as the figure below?
I
A.
I and II only
B.
III only
C.
III and IV only
D.
I, II, III and IV
II
III
IV
9. In the figure, VABCD is a right pyramid with a square base. If M, N, O and P are the mid-points of AB,
BC,CD and DA respectively, which of the following are right angles?
V
I. VXN
II. NXO
III. MXD
IV. BAC
A.
I and II only
B.
II and IV only
C.
I, II and IV only
D.
I, II, III and IV
O
D
C
P
N
X
A
B
M
10. In the figure, VABCD is a right pyramid with a rectangular base. Which of the following is the angle
between the line segment VC and the base ABCD?
V
A.BCD
B.VCB
C.VCD
C
B
D.VCA
D
A
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11.In the figure, ABCDEFGH is a cuboid. Which of the following is the angle between the plane GFDC and
the plane EDCH?
E
F
A. FDC
G
B.FDE
H
C.GCD
A
D
D.GCB
B
C
12. The figures shows the cube ABCDEFGH of side 2 cm. X and Y are the mid-points of AB and GH respectively.
Find XY.
(2004CE)
C
D
A. 3 cm
B. 2 2 cm
C.
5 cm
D.
6 cm
A
B
X
E
H
Y
F
G
13. In the figure, ABCDEFGH is a rectangular block. EG and FH meet at X. M is the mid-point of EH. Which of the
following makes the greatest angle with the plane ABCD?
(2004CE)
M
E
H
A. AG
X
F
B. AH
G
C. AM
D. AX
10 cm
D
14. The figure shows a cuboid. Which of the following are right angles?
I. CAF
II. DHG
III. AGC
A.
B.
C.
D.
C
D
A
I & II only
I & III only
II & III only
I, II & III
B
E
F
H
G
C
6 cm
A
8 cm B
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Level 3 (Optional)
1.
The figure shows a rectangular prism with sides
of lengths 9cm, 5cm and 4cm.
Calculate the angles between the diagonal EC and
(a) the plane ABCD,
F
(b) the plane ADHE,
(c) the plane AEFB.
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E
H
A
4cm
G
D
5cm
B
9cm
C
Triangles to be considered in each part.
(a)
E
A
5cm
4cm
B
9cm
C
A
(b)
C
E
E
4cm
D
5cm
9cm
D
(c) A
E
E
C
4cm
5cm
B
B
9cm
A
C
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2.
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EABCD is a right pyramid with slant edges EA, EB, EC, ED, are equal.
The base ABCD is a square with AB = 12cm. The height EF = 20cm.
J and H are the mid-points of AD and AB respectively.
(a) Find the length of HF and BF.
E
(b) Find the angle between the line EH and the base ABCD.
Hence find the angle between the line EJ and the base ABCD.
(c) Find the angle that EB makes with the base ABCD.
Hence find also the angle that EA makes with
the base ABCD.
20cm
C
B
F
D
H
J
A
Triangles to be considered in each part.
(a)
B
H
F
(b)
E
20cm
(c)
F
H
E
20cm
F
B
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3.
ABCD and CDEF are two rectangular planes
perpendicular to each other. If FBC＝300
E
and FAB＝600, FC＝xcm.
(a) Add suitable straight line to the diagram
and mark the angle between the straight
line AF and the plane ABCD.
D
(b) Find, in terms of x, the lengths of
(i) BF,
(ii) AF
600
(c) Find the angle between the line AF and
A
the plane ABCD correct to the nearest degree.
3/2006 P.15
F
x cm
C
0
30
B
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Reading
How Do We Know There Are Only Five?
First of all, we need to know the vertex angles for a regular polygon of n sides. We can draw radii from the
center to divide the polygon into n triangles. The total of all the angles of all the triangles is thus 180n. The
angles at the center of the polygon add up, of course, to 360, so the angles around the edge of the
polygon must total 180n-360 or 180(n-2). Each vertex angle, then, equals 180(n-2)/n. Thus the angle of
an equilateral triangle is 60 degrees (180(3-1)/3), a square is 90 degrees (180(4-2)/4), a pentagon is 108
and a hexagon is 120.
There have to be at least three polygons at a vertex; two would simply fold together back to back. Also the
angles around a vertex cannot exceed 360 degrees, and if they equal 360 degrees, the polygons simply
tile the plane. Thus we can rule out solids with 6 or more faces - hexagons tile the plane and all other
triplets exceed 360 degrees. That means convex solids with regular polygon faces can only have triangles,
squares or pentagons as faces. The only possibilities are 3, 4, or 5 triangles, 3 squares or 3 pentagons.
Thus there are only five platonic solids.
Edges
Solid
Faces of
Face
Tetrahedron
4
3
Cube
6
4
Octahedron
8
3
Dodecahedron 12
5
Icosahedron 20
3
Edges
Vertices at
Vertex
4
3
8
3
6
4
20
3
12
5
Edges
6
12
12
30
30
Note that in each case, Euler's Rule is followed: F + V = E + 2. Also note that faces in the cube and
vertices in the octahedron play similar roles, and similarly for the dodecahedron and icosahedron. These
solids are duals of each other.
Platonic Solids and Tilings
Platonic solids and uniform tilings are closely related as shown below. Starting from the tetrahedron we
have polyhedra with three, four and five triangles at each vertex, then the plane tiling with six triangles
(top row). Also starting from the tetrahedron we have polyhedra with three triangles, squares and
pentagons at
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each vertex, then the plane tiling with
three hexagons (bottom row). The figures
in the top and bottom rows are duals of
each other; the tetrahedron is its own dual.
The square tiling is placed between the cube
and octahedron. We can regard it as the next
step from the octahedron (four polygons at
a vertex) or from the cube (three and four
squares at a vertex). The square tiling is its
own dual.
History of Plato Solids
The Greeks, who were inclined to see in mathematics something of the nature of religious truth,
found this business of there being exactly five Platonic solids very compelling. The philosopher
Plato concluded that they must be the fundamental building blocks – the atoms – of nature, and
assigned to them what he believed to be the essential elements of the universe. He followed the
earlier philosopher Empedocles in assigning fire to the tetrahedron, earth to the cube, air to the
octahedron, and water to the icosahedron. To the dodecahedron Plato assigned the element
cosmos, reasoning that, since it was so different from the others in virtue of its pentagonal faces,
it must be what the stars and planets are made of.
Although this might seem naive to us, we should be careful not to smile at it too much:
these were powerful ideas, and led to real knowledge.
As late as the 16th century, for instance, Johannes Kepler was applying a
similar intuition to attempt to explain the motion of the planets. Early in
his life he concluded that the distances of the orbits, which he assumed
were circular, were related to the Platonic solids in their proportions. This
model is represented in this woodcut from his treatise Mysterium
Cosmographicum.
The beauty and interest of the Platonic solids continue to inspire all
sorts of people, and not just mathematicians.
索瑪立方塊(Soma Cube)
由七片組件的總體積為 27 單位，可以重拼成 3x3x3 的正立方體。