21
ES
Ch apter
PA
G
Measurement and Geometry
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AL
Polyhedra and three-­
dimensional drawing
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In the real world we have to deal with three-dimensional objects. When we
represent these objects in drawings, we draw on paper and this allows us to
work only in two dimensions. This is a significant challenge, as real objects can
be viewed from many different directions and look quite different from each
direction. If a viewer looks at a three-dimensional object from three different
directions, an understanding of the shape of the object is achieved. A drawing
on paper, on the other hand, can represent this object only as viewed from one
particular direction. No matter how much we move to view the drawing from
different directions, it does not change our understanding of the drawing.
The difficulty of drawing three-dimensional objects is exemplified by the fact
that it is only about 500 years ago that painters first mastered the techniques of
perspective that made their paintings in the real world look three dimensional.
Drawing simple three-dimensional shapes is an important skill for a student of
mathematics. In this chapter we consider some basic techniques and apply them
to some three-dimensional geometric shapes.
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21 A
Polyhedra
A polyhedron is a solid where the outside surface is made up wholly of flat panels called faces
with simple polygonal shapes and is not pierced by holes. We note that the plural of polyhedron is
polyhedra or polyhedrons. A polygonal shape is a closed plane shape with a border made up of
intervals. Triangles and quadrilaterals are examples of polygonal shapes. So, solids like spheres,
tubes and living things are not polyhedra, but there are many manufactured solids such as boxes,
houses and books that are very close to being ideal mathematical polyhedra.
Platonic solids
Truncated cube
PA
Pentagonal prism
G
ES
Here are three polyhedra. The two to the left are named and the third described.
Cube with square pyramid
on top of it
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N
AL
We recall from Chapter 18 that a regular polygon is a plane figure with sides that are intervals of
the same length. A regular polyhedron is a three-dimensional solid with faces that are all identical
regular polygons and with the same number of edges meeting at each vertex. There are only five
such solids and they are shown below. They are also known as the Platonic solids. They have many
interesting properties.
Tetrahedron
Cube
Icosahedron
Octahedron
Dodecahedron
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2 1 A P o ly h e d r a
The following are not polyhedrons since their faces are not polygons.
Sphere
Cylinder
Cone
Names for parts of a polyhedron
The pyramid shown opposite has five faces.
Two faces of a polyhedron meet along a line called an edge.
The pyramid shown opposite has eight edges.
face
The point at which three or more edges meet is called a vertex.
Prisms
edge
G
The pyramid shown opposite has five vertices.
vertex
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Each polygon forming the surface of a polyhedron is called a face.
N
AL
PA
A prism is a polyhedron that has two identical and parallel faces and all of its remaining faces are
parallelograms. The following are examples of prisms.
Cross-section of a prism
A cross-section of a prism is the polygon produced by intersecting the prism by a plane parallel to
the base.
FI
If the intersecting plane is parallel to the planes containing the identical faces, the cross-section will
be identical to the faces at the ends and so, a prism has a uniform cross-section.
For example, for the prism shown to the left below, the cross-section is always a triangle that is
indentical to the triangle that is the base. The figure is called a right-triangular prism.
The figure to the right is called a right-rectangular prism.
Oblique prisms
The figure shown to the right is an oblique rectangular prism.
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2 1 A P o ly h e d r a
Exercise 21A
1 Determine which of the following shapes are polyhedra.
a
b
e
f
ES
d
c
e
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AL
d
PA
G
2 Find the number of faces, edges and vertices of the following solids.
a
b
c
FI
3 Which of the following are prisms?
a
b
d
e
c
f
g
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4 a Complete the following table for prisms with the given cross-sections.
Triangular prism
Rectangular prism
Pentagonal prism
Triangular
Rectangular
Pentagonal
G
Hexagonal
b For each of the above, evaluate V − E + F
Number of vertices (V)
ES
Type of cross-section Number of faces (F) Number of edges (E)
Hexagonal prism
i the number of faces (F)
ii the number of edges (E)
PA
c If the polygonal cross-section of a prism has n sides, find in terms of n:
iii the number of vertices (V)
d Find V − E + F using the results of parts i, ii and iii.
FI
N
AL
5 If a pyramid has a triangular base it is called a triangular pyramid. If it has a pentagonal
base it is called a pentagonal pyramid. Several pyramids are illustrated below.
Triangular pyramid
Rectangular pyramid
Pentagonal pyramid
Hexagonal pyramid
a Complete the table.
Type of pyramid
Number of faces (F)
Number of edges (E)
Number of vertices (V)
Triangular
Rectangular
Pentagonal
Hexagonal
b For each of the above evaluate V − E + F.
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c If the polygonal base has n sides, find in terms of n:
i the number of faces (F)
ii the number of edges (E)
iii the number of vertices (V)
d Find V − E + F using the results of parts i, ii and iii.
6 aComplete the following table. Diagrams of the platonic solids are on page 523. It is a
good idea to have models of these solids when doing this question.
Number of faces (F)
Number of edges (E)
Cube
Tetrahedron
Icosahedron
Octahedron
Dodecahedron
b For each of the above evaluate V − E + F.
Number of vertices (V)
ES
Platonic solid
Drawing a solid
N
AL
21 B
PA
G
In Questions 4, 5 and 6 you calculated V − E + F. You should have got 2 in each case.
The formula V − E + F = 2 is known as Euler’s formula.
Views or elevations
The most basic drawings of a solid are done according to the following rules.
The front view (or front elevation) outlines what is seen as you stand directly in front of the solid.
The rear view (or rear elevation) outlines what is seen as you stand directly behind the solid.
FI
A side view (or side elevation) outlines what is seen as you look at the solid from a side. This line of
view should be at right angles to the front view.
The plan view outlines what is seen when the solid is observed from directly above.
House plans are often presented in this way. The front and one side view of a house are shown here.
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Three views of a house are shown below.
Front view Side view Plan view
G
Although none of these drawings by itself can give a good representation
of the solid, we have the ability to use the three drawings to produce
an image of the solid in our mind.
Isometric drawing
PA
An isometric drawing combines the three views into one. The object
is drawn so that the front, side and top views of the solid are all visible
to the person making the drawing. This drawing is called the isometric
drawing of the solid. The isometric drawing for the house in the
previous section is shown.
N
AL
It should be noted that, from this view, the geometrical properties of the
faces of the solid are no longer maintained. The right angles are not right angles when measured.
However, parallel lines in the original solid remain parallel in the isometric drawing.
Drawing on isometric paper
FI
Isometric paper is a type of paper used for drawing solids from an isometric viewpoint. The angles
formed by the lines on the paper represent right angles from the isometric viewpoint, so this paper
is best suited for drawing shapes with horizontal bases and side faces that are upright and at right
angles to each other. Isometric paper is available to download from your Interactive Textbook. The
simplest solid with these properties is a cube. The diagrams below show two cubes of different sizes
and a short set of steps drawn on isometric paper.
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2 1 B D r a w ing a s o l i d
On the right are the three different ways two cubes
stuck together to form one prism may be drawn on
isometric paper.
Example 1
G
ES
Draw the front view, right side view and plan view of the solid.
N
AL
Solution
PA
right side
front
Plan view
Right side view
FI
Front view
Exercise 21B
For this exercise it is useful to have a set of unit cube blocks.
1 Draw each of the following on isometric paper.
a
b
c
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2 1 B D r a w ing a s o l i d
d
Example 1
e
f
2 Draw the front, side and plan views of the following solids.
a
b
front
c
side
ES
side
e
f
G
d
front
front
side
side
PA
front
side
front
side
front
N
AL
3 a On isometric paper, draw three cubes, all on the same level and stuck together to form a
prism, in as many ways as you can. You will find that there are only two such prisms but
different isometric drawings of each of them.
b I s it possible to make an object that is not a prism if the three cubes are stuck together
such that each cube has at least one face stuck to the face of another cube?
FI
4 a Draw four other objects that can be made
from four cubes stuck together as described in
Question 3b. Only one version of each object
needs to be drawn on isometric paper. Three of
them have been drawn for you here.
b How many of these objects were not prisms?
5 Do two other isometric drawings of each of the following solids.
a
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b
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Review exercise
1 A wedge-shaped solid is shown. It is drawn on isometric dot paper.
B
C
E
ES
D
A
F
b Name the faces of this prism.
G
a What type of prism is this?
c Draw the plan view, back view and side view of this solid.
PA
d Draw the prism again from a different view on isometric dot paper.
e Verify that V − E + F = 2.
N
AL
2 a For the figure shown, draw the plan view, front view, back view and side view.
FI
side
front
b Is the figure a prism?
3 Draw two different isometric drawings of each of the following.
a
b
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Challenge exercise
In the following work with Euler’s formula, V − E + F = 2.
1 Find a polyhedron with the given number of edges and vertices. First find the number of
faces using Euler’s formula.
a 8 edges, 5 vertices
b 9 edges, 6 vertices
c 16 edges, 8 vertices
2 A tetrahedron has four faces and four vertices. Name a polyhedron with:
b six vertices and six faces
ES
a five vertices and five faces
PA
G
3 The diagram is of a solid called an extended triangular pyramid. It has seven faces and
seven vertices.
a Draw an extended square pyramid.
b How many faces, edges and vertices does an extended square pyramid have?
N
AL
c A pyramid has a polygonal base with n sides. How many vertices and faces does the
associated extended pyramid have? How many edges does it have?
d Name two polyhedra that have nine faces and nine vertices.
4 aDraw a prism with a cross-section that is a quadrilateral with two parallel sides of
unequal length and the other two sides of equal length.
b Show with a drawing how two of the prisms of part a may be put together to form a
hexagonal prism.
FI
c Show with a drawing how a number of prisms with cross-section a regular hexagon
may be put together to form a ‘honeycomb’.
5 aDraw two tetrahedrons joined at a base. Describe the polyhedron you get. Calculate
V − E + F for this polyhedron.
b A parallelepiped is a prism with all faces a parallelogram. Draw a parallelepiped.
c Show how a tetrahedron can be inscribed in a parallelepiped with each edge of the
tetrahedron lying on a face of the parallelepiped.
d Find, by means of a rough sketch, what polyhedron has its vertices at the midpoints of
the edges of a tetrahedron.
6 Show with a drawing how a cube can be cut up into six identical square pyramids.
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