More about 3-D Figures

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More about 3-D Figures
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Case Study
4.1 Symmetry of 3-D Figures
4.2 Nets of 3-D Figures
4.3 2-D Representations of 3-D Figures
4.4 Points, Lines and Planes of 3-D Figures
4.5 Further Exploration on 3-D Figures
Chapter Summary
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Case Study
We havent got enough boxes for packing the goods.
Can you help them to make a box using cardboard?
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Case Study
As shown in the figure below, imagine that we cut
a box along the orange edges and unfold the box.
If we copy the unfolded plane figure onto a piece
of cardboard and reverse the above procedure, we
can fold the piece of cardboard into a box.
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4.1 Symmetry of 3-D Figures
A. Reflectional Symmetry
(a) Introduction A 3-D object is said to have
reflectional symmetry if a plane can divide the
object into 2 identical parts which are mirror
images of each other.
Such a plane is called the plane of reflection.
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4.1 Symmetry of 3-D Figures
A. Reflectional Symmetry
Some objects have more than one plane of
reflection.
The figure below shows 3 of the 4 planes of
reflection of a regular triangular prism.
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4.1 Symmetry of 3-D Figures
A. Reflectional Symmetry
(b) Cube A cube is a regular solid with 6
identical squares as faces.
There are 9 different planes of reflection in a
cube.
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4.1 Symmetry of 3-D Figures
A. Reflectional Symmetry
(c) Regular Tetrahedron A regular tetrahedron is
a solid with 4 identical equilateral triangles as
faces.
There are 6 different planes of reflection in a
regular tetrahedron.
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B. Rotational Symmetry
4.1 Symmetry of 3-D Figures

• Introduction
• A 3-D figure is said to have rotational symmetry
  if it repeats itself more than once when it is
  rotated about a line in one complete revolution.
  Such a line is called the axis of rotational
  symmetry.

The above cuboid has 2-fold rotational symmetry
or rotational symmetry of order 2 about the line
PQ.
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B. Rotational Symmetry
4.1 Symmetry of 3-D Figures
Some objects have more than one axis rotational
symmetry.
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B. Rotational Symmetry
4.1 Symmetry of 3-D Figures
(b) Cube There are 13 different axes of
rotational symmetry in a cube.
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B. Rotational Symmetry
4.1 Symmetry of 3-D Figures
(c) Regular Tetrahedron There are 7 different
axes of rotational symmetry in a regular
tetrahedron.
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4.2 Nets of 3-D Figures
A net is a 2-D pattern which can be folded into a
solid.
1. There are many different nets for a cube.
2. Each edge will coincide with exactly one other
edge of the net to form a solid.
3. There are totally 11 different nets for a cube.
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Example 4.1T
4.2 Nets of 3-D Figures
The figure shows an open box. Draw a net of the
box.
Solution
If we cut along the sides and unfold the box, we
can obtain its net.
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4.2 Nets of 3-D Figures
Example 4.2T
The figure shows a net of a cube. When it is
folded into a cube, what is the face opposite to
face A?
Solution
The face opposite to face A is face E.
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4.3 2-D Representations of 3-D Figures
A. 3-D Objects in 2-D Views
If we look at an object from different
directions, we can obtain different views.
For example, the following shows different views
of an object below. They are the front, top and
side views.
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4.3 2-D Representations of 3-D Figures
A. 3-D Objects in 2-D Views
Example 4.3T
Draw the front view, top view and side view of
the solid shown.
Solution
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B. Sketch 3-D Objects from 2-D Views
4.3 2-D Representations of 3-D Figures
We can sketch 3-D objects from their 2-D views.
When sketching the solid, we may
1. first sketch the solid from the front view,
2. next sketch the solid according to the top
view and the side view,
3. then check the ratio of the lengths of the
sides of the solid.
4. Lastly, find the 3 views from the sketch to
check whether they match the given 2-D
information.
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4.3 2-D Representations of 3-D Figures
B. Sketch 3-D Objects from 2-D Views
Example 4.4T
The following shows the front view, top view and
side view of a solid. Sketch the solid.
Solution
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C. Limitation of Plane Figures
4.3 2-D Representations of 3-D Figures
There is a limitation in using 2-D representation
of a solid to figure out the actual solid.
For example, for the front view and side view as
shown, we can obtain many possible solids, such
as
Therefore, the more the views of an object that
are given, the better the shape of the solid we
can obtain.
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4.4 Points, Lines and Planes of 3-D Figures
In 2-dimensional space, we learnt that
1. 2 non-parallel lines L1 and L2 would intersect
at a point O with an angle q as shown in the
figure,
2. the perpendicular distance between a point P
and a line L3 is their shortest distance, denoted
by PQ as shown in the figure.
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A. Angle between a Line and a Plane
4.4 Points, Lines and Planes of 3-D Figures
(a) Projection of a Point on a Plane In the
figure, O is a point on the plane p and V is a
point not on the plane.
If VO is perpendicular to any lines on the plane
(say L1 and L2), then O is called the projection
of point V on the plane.
VO is the shortest distance between point V and
the plane.
In the top view of the figure, point V coincides
with its projection O.
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A. Angle between a Line and a Plane
4.4 Points, Lines and Planes of 3-D Figures
(b) Angle between a Line and a Plane In the
figure, A is a point on the plane p and the line
AB does not lie on the plane.
If C is the projection of B on the plane, then
AC is the projection of AB on the plane.
q is the angle between AB and the plane.
1. In the top view of the figure, line
AB coincides with its projection AC.
2. If a line not on the plane is parallel
to the plane, then we cannot find the
point of intersection.
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4.4 Points, Lines and Planes of 3-D Figures
A. Angle between a Line and a Plane
Example 4.5T
The figure shows a cube ABCDHEFG. Write down the
angles between line DF and the following
planes. (a) Plane CDHG (b) Plane BCGF
Solution
(a) ? G is the projection of F on plane CDHG.
? The angle between line DF and plane CDHG is
?FDG.
(b) ? C is the projection of D on plane BCGF.
? The angle between line DF and plane BCGF is
?DFC.
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B. Angle between Two Planes
4.4 Points, Lines and Planes of 3-D Figures
In the figure, 2 non-parallel planes p1 and p2
intersect and they meet at a line AB which is
called the line of intersection.
To find the angle between 2 planes p1and p2,
1. first construct a perpendicular line PQ of AB
on plane p1, where Q is a point on AB,
2. then through Q, construct a perpendicular line
QR of AB on plane p2.
3. As a result, the required angle is ?PQR.
2 planes are parallel if they do not intersect
even after they are extended.
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4.4 Points, Lines and Planes of 3-D Figures
B. Angle between Two Planes
Example 4.6T
The figure shows a cuboid ABCDHEFG. Write down
the angles between the following planes. (a)
Planes ADHE and EFGH (b) Planes ADGF and BCGF
Solution
(a) ? EH is the line of intersection of the 2
planes.
? The angle between planes ADHE and EFGH is
?DHG or ?AEF.
(b) ? FG is the line of intersection of the 2
planes.
? The angle between planes ADGF and BCGF is
?DGC or ?AFB.
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4.4 Points, Lines and Planes of 3-D Figures
B. Angle between Two Planes
Example 4.7T
The figure shows a regular tetrahedron ABCD. P,
Q, R, S, T and U are the mid-points of the edges.
Write down the angles between (a) planes PCD and
BCD, (b) planes ABD and ACD.
Solution
(a) ? CD is the line of intersection of the 2
planes, and PT and BT are perpendicular to CD.
? The angle between planes PCD and BCD is ?PTB.
(b) ? AD is the line of intersection of the 2
planes, and CR and BR are perpendicular to AD.
? The angle between planes ABD and ACD is ?CRB.
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4.5 Further Exploration on 3-D Figures
A. Eulers Formula
The relationship between the numbers of vertices,
edges and faces of a polyhedron is as follows.
V ? E ? F ? 2
This is called Eulers formula, where V is the
number of vertices, E is the number of edges and
F is the number of faces.
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4.5 Further Exploration on 3-D Figures
A. Eulers Formula
Example 4.8T
The figure shows the net of a polyhedron. (a) What
polyhedron can the net be folded into? (b) Find
the number of vertices (V), the number of edges
(E) and the number of faces (F) of the
polyhedron. (c) Does the Eulers formula hold?
Solution
(a) Triangular prism
(b) V ? 6, E ? 9, F ? 5
(c) V ? E ? F ? 6 ? 9 ? 5
? 2
? Eulers formula holds.
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B. Duality of Regular Polyhedra
4.5 Further Exploration on 3-D Figures
In Book 1B Chapter 8, we learnt that there are 5
regular polyhedra
When the number of vertices and the number of
faces of 2 polyhedra are reversed, the 2
polyhedra are called dual polyhedra.
For example
Regular Hexahedron V ? 8, E ? 12, F ? 6
Regular Octahedron V ? 6, E ? 12, F ? 8
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B. Duality of Regular Polyhedra
4.5 Further Exploration on 3-D Figures
For a dual pair, each vertex of a polyhedron
touches the mid-point of one of the faces of the
other polyhedron as shown in the following figure.
Apart from regular hexahedron and regular
octahedron, can you find another dual pair?
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Chapter Summary
4.1 Symmetry of 3-D Figures
1. Reflectional Symmetry A 3-D figure is said
to have reflectional symmetry if a plane, that is
the plane of reflection, can divide the figure
into 2 identical parts which are mirror images of
each other.
The figure shows a plane of reflection of a
binder clip.
2. Rotational Symmetry A 3-D figure is said to
have rotational symmetry if it repeats itself
more than once when it is rotated about a line,
that is the axis of rotational symmetry, in one
complete revolution.
The figure shows an axis of rotational symmetry
of an hourglass.
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Chapter Summary
4.2 Nets of 3-D Figures
1. A net is a 2-D pattern which can be folded
into a solid.
2. There may be different nets for the same 3-D
figure.
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Chapter Summary
4.3 2-D Representations of 3-D Figures
1. If we look at an object from different
directions, we can obtain different views.
2. We can identify 3-D objects from their 2-D
views.
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Chapter Summary
4.4 Points, Lines and Planes of 3-D Figures
1. If VO is perpendicular to any lines (say L1
and L2) on a plane, then O is called the
projection of V on the plane.
2. If C is the projection of B on a plane, then
AC is the projection of AB on the plane, and q
is the angle between AB and the plane.
3. ?PQR is the angle between 2 planes p1 and p2,
where AB is the line of intersection of the
planes, PQ ? AB and QR ? AB.
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Chapter Summary
4.5 Further Exploration on 3-D Figures
1. Eulers Formula For a polyhedron, V ? E ? F ?
2, where V is the number of vertices, E is the
number of edges and F is the number of faces.
2. Duality of Regular Polyhedra For dual
polyhedra, each vertex of one of the polyhedron
touches the mid-point of one of the faces of the
other polyhedron.
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4.2 Nets of 3-D Figures
Follow-up 4.1
The figure shows an open box. Draw a net of the
box.
Solution
If we cut along the sides and unfold the box, we
can obtain its net.
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4.2 Nets of 3-D Figures
Follow-up 4.2
The figure shows a net of a cube. When it is
folded into a cube, what is the face opposite to
face D?
Solution
The face opposite to face D is face A.
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4.3 2-D Representations of 3-D Figures
A. 3-D Objects in 2-D Views
Follow-up 4.3
Draw the front view, top view and side view of
the solid shown.
Solution
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4.3 2-D Representations of 3-D Figures
B. Sketch 3-D Objects from 2-D Views
Follow-up 4.4
The following shows the front view, top view and
side view of a solid. Sketch the solid.
Solution
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4.4 Points, Lines and Planes of 3-D Figures
A. Angle between a Line and a Plane
Follow-up 4.5
The figure shows a cube ABCDHEFG. Write down the
angles between line BG and the following
planes. (a) Plane EFGH (b) Plane ABFE
Solution
(a) ? F is the projection of B on plane EFGH.
? The angle between line BG and plane EFGH is
?BGF.
(b) ? F is the projection of G on plane ABFE.
? The angle between line BG and plane ABFE is
?GBF.
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4.4 Points, Lines and Planes of 3-D Figures
B. Angle between Two Planes
Follow-up 4.6
The figure shows a cuboid ABCDHEFG. Write down
the angles between the following planes. (a)
Planes ABCD and BCGF (b) Planes BCHE and ADHE
Solution
(a) ? BC is the line of intersection of the 2
planes.
? The angle between planes ABCD and BCGF is
?ABF or ?DCG.
(b) ? EH is the line of intersection of the 2
planes.
? The angle between planes BCHE and ADHE is
?CHD or ?BEA.
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4.4 Points, Lines and Planes of 3-D Figures
B. Angle between Two Planes
Follow-up 4.7
The figure shows a square pyramid VABCD. O is the
point of intersection of the diagonals of square
ABCD and it is the projection of V on the plane
ABCD. P and Q are the mid-points of BC and AD
respectively. Write down the angles between (a)
planes VBC and ABCD, (b) planes VBC and VAD.
Solution
(a) ? BC is the line of intersection of the 2
planes, and VP and PQ are perpendicular to BC.
? The angle between planes VBC and ABCD is
?VPO.
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4.4 Points, Lines and Planes of 3-D Figures
B. Angle between Two Planes
Follow-up 4.7
The figure shows a square pyramid VABCD. O is the
point of intersection of the diagonals of square
ABCD and it is the projection of V on the plane
ABCD. P and Q are the mid-points of BC and AD
respectively. Write down the angles between (a)
planes VBC and ABCD, (b) planes VBC and VAD.
Solution
(b) Consider a line VX passing through V
and parallel to plane ABCD.
? VX is the line of intersection of the 2
planes, and VP and VQ are perpendicular to VX.
? The angle between planes VBC and VAD is ?PVQ.
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4.5 Further Exploration on 3-D Figures
A. Eulers Formula
Follow-up 4.8
The figure shows the net of a polyhedron. (a) What
polyhedron can the net be folded into? (b) Find
the number of vertices (V), the number of edges
(E) and the number of faces (F) of the
polyhedron. (c) Does the Eulers formula hold?
Solution
(a) Pentagonal prism
(b) V ? 10, E ? 15, F ? 7
(c) V ? E ? F ? 10 ? 15 ? 7
? 2
? Eulers formula holds.