Polygons and
polyhedra
11
Val and Peter want to
replace their front gate with
another of the same design.
What shapes are formed by
the metal bars of the gate?
To have this gate made,
they need to supply a
diagram of it with all
measurements and angles
shown. In this chapter, you
will look at different shapes
and their properties,
including angles.
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Maths Quest 7 for Victoria
Introduction
The world around you is filled with many different shapes and objects. The roof of this
house in Falls Creek is shaped as a triangle when viewed from the front. This lets the
winter snow slide off the roof.
Dice are shaped as cubes so that each
of the 6 numbers are equally likely to
appear on the uppermost face.
This soccer ball is made up of five-sided and
six-sided shapes that almost form a sphere
which can be rolled and kicked along the
ground.
As you learn about the properties of
shapes and objects you will understand
how and why they are used in the world
around you.
In this chapter you will learn the
names and properties of many common
2-dimensional shapes, called polygons, and
3-dimensional objects, called polyhedra. You
will also learn to construct 3-dimensional
objects and draw them on a page.
Chapter 11 Polygons and polyhedra
423
Types of triangles
The word triangle means 3 angles. Every triangle has 3 angles and 3 sides.
Capital letters of the English alphabet at the
vertices of triangles can be used to identify
B
triangles. When identifying triangles, the vertices
are listed in either a clockwise or anticlockwise
direction, beginning with any vertex. Instead of
the word ‘triangle’ the symbol L is used.
For example, the triangle shown at right can be
A
C
referred to as LABC. However, it would be
equally appropriate to name it LBCA, LCAB,
LACB, LBAC or LCBA.
Triangles can be classified according to the length of their sides or the size of their
angles.
Classifying triangles according to the length of their
sides
An equilateral triangle has all sides equal in length.
Note that identical marks on the sides of a triangle are used
to indicate that the sides have the same length. The angles of
an equilateral triangle are equal in size. This is shown by
placing identical curves on each angle.
An isosceles triangle has 2 sides of equal length.
The side of the isosceles triangle that has a different length,
is often called the base of the triangle. The angles adjacent to
the base of the isosceles triangle are equal in size. On the
diagram at right, the side markings show the 2 sides that are
equal and the angle markings show the 2 angles that are equal.
A scalene triangle has no equal sides.
The different side markings on the diagram show that
the 3 sides have different lengths. A scalene triangle has
all 3 angles of different size. This is shown by different
angle markings.
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Maths Quest 7 for Victoria
WORKED Example 1
Classify each of these triangles according to the lengths of their sides.
a
b
c
N
R
B
S
A
C
M
P
P
THINK
WRITE
a Sides AB and AC have identical
markings on them, which indicates that
they are of equal length. So LABC has
2 equal sides. Classify it accordingly.
a LABC is an isosceles triangle.
b The 3 sides of LMNP have identical
markings on them, which means that all
3 sides are equal in length. Classify this
triangle.
b LMNP is an equilateral triangle.
c All 3 sides of LPRS are marked
differently. Therefore, no sides in this
triangle are equal in length. Use this
information to classify the triangle.
c LPRS is a scalene triangle.
Classifying triangles according to the size of their angles
A right-angled triangle has one of its angles equal to 90°
(that is, one of its angles is a right angle).
On the diagram, putting a small square in the corner marks the right
angle.
An acute-angled triangle has all angles smaller than 90°
(that is, all 3 angles are acute).
An obtuse-angled triangle has 1 angle greater than 90°
(that is, one angle is obtuse).
Chapter 11 Polygons and polyhedra
425
WORKED Example 2
Classify each of the triangles in worked example 1 according to the size of their angles.
THINK
WRITE
a In LABC, ∠CAB is marked as the right
angle, so classify it accordingly.
a LABC is a right-angled triangle.
b In LMNP all angles are less than 90°,
so classify this triangle.
b LMNP is an acute-angled triangle.
c In LPRS, ∠PRS is greater than 90°;
that is, it is obtuse. Use this information
to classify the triangle.
c LPRS is an obtuse-angled triangle.
remember
remember
1. According to the lengths of the sides, a triangle can be classified as being:
(a) equilateral (3 equal sides)
(b) isosceles (2 equal sides)
(c) scalene (no equal sides).
2. A triangle can be classified according to the angle size, as being:
(a) acute-angled (all 3 angles are acute)
(b) right-angled (1 angle is a right angle)
(c) obtuse-angled (1 angle is obtuse).
11A
WORKED
Example
1
Types of triangles
1 Classify each of these triangles according to the lengths of their sides.
a
b
c
d
e
f
11.1
Classifying
triangles
(sides)
426
WORKED
Example
Classifying
triangles
(angles)
Maths Quest 7 for Victoria
2 Classify each of the triangles in question 1 according to the size of their angles.
2
3 Add side and angle markings to these diagrams to show that:
V
a
b
S
U
R
T
LRST is an equilateral triangle
c
P
LUVW is an isosceles triangle
d
Q
N
M
R
LPQR is a scalene triangle
e
W
P
LMNP is a right-angled triangle
f
B
M
N
O
A
C
LABC is a right-angled and
isosceles triangle
LMNO is a right-angled and
scalene triangle.
4 multiple choice
a Which of these triangles is an equilateral triangle?
A
B
D
E
C
Chapter 11 Polygons and polyhedra
b Which of these triangles is not a scalene triangle?
A
B
D
E
C
5 multiple choice
a Which of these triangles is both right-angled and scalene?
A
B
D
E
C
b Which of these triangles is both acute-angled and isosceles?
A
B
D
E
C
427
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Maths Quest 7 for Victoria
6 What types of triangles can you see
in this picture?
7 Write down 3 acute triangles you can
see around you.
8 Find one example in your classroom
or home of each of the 6 types of
triangles described in this chapter.
Describe clearly where the triangle
occurs, draw the triangle and classify
it according to both side and angle
types.
9 In the picture at right:
a how many equilateral triangles can you find?
b how many right-angled triangles can you find?
c how many isosceles triangles can you find?
QUEST
GE
S
EN
MAT H
10 Use your ruler, pencil and protractor to accurately draw:
a an equilateral triangle with side lengths 6 cm and all angles 60°
b an isosceles triangle with two sides which are 6 cm each with a 40° angle between
them
c a right-angled triangle whose two short sides are 6 cm and 8 cm.
How long is the longer side?
d a scalene triangle with two of the sides measuring 4 cm and 5 cm and an angle of
70° between the two sides.
CH
AL
1 How many triangles can you find in these shapes?
a
b
L
2 How many triangles can you find in these shapes?
a
b
Chapter 11 Polygons and polyhedra
429
Angles in a triangle
Sum of angles in a triangle
You will need: a ruler and a protractor.
1. Draw an acute-angled triangle in your workbook.
2. Use a protractor to measure each of the 3 angles.
3. Find the sum of the 3 angles.
4. Draw up a table like the one shown below and write in your results.
Triangle
First angle
Second
angle
Third angle
Sum of
angles
1. Acute-angled
2. Obtuse-angled
3. Right-angled
4. Isosceles
5. Scalene
5. Repeat steps 1–3 for the 4 other triangles in the table.
Angle sum
of a
triangle
6. Write down any patterns that you have observed in relation to the sum of the
angles in a triangle.
As a result of this investigation you should have discovered the rule that is stated below.
It can be shown that the sum of the 3 angles in any
triangle is equal to 180°.
In the triangle at right, a + b + c = 180° where the
3 angles of the triangle are a, b and c
b
a
c
This rule can be used to find missing angles in triangles, as shown in the following
examples.
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Maths Quest 7 for Victoria
WORKED Example 3
Find the value of the pronumeral in this triangle.
35º
THINK
1
2
3
The sum of the 3 angles (b, 35° and
58°) must be 180°. Write this as an
equation.
Simplify by adding 35° and 58°
together.
Use inspection or backtracking to solve
for b.
58º
b
WRITE
b + 35° + 58° = 180°
b + 93° = 180°
b = 180° − 93°
b = 87°
+ 93°
b
b + 93°
87°
180°
– 93°
In the previous section it was discussed that the angles at the base of an
isosceles triangle are equal in size. Worked examples 4 and 5 illustrate the use of
this property.
WORKED Example 4
Find the value of the pronumeral in the following triangle.
B
h
A
74º
THINK
1
2
3
4
The markings on the diagram indicate
that LABC is isosceles with AB = BC.
Therefore, the angles at the base are
equal in size; that is, ∠BCA = ∠BAC =
74°.
All 3 angles in a triangle must add up to
180°.
Simplify.
Solve for h.
WRITE
C
∠BAC = 74°
∠ABC + ∠BAC + ∠BCA = 180°
h + 74° + 74° = 180°
h + 148° = 180°
h = 180° − 148°
h = 32°
Chapter 11 Polygons and polyhedra
431
WORKED Example 5
Find the value of the pronumeral in the following triangle.
N
40º
a
M
THINK
P
WRITE
1
From the diagram we can see that
LMNP is isosceles with MN = NP.
Hence, ∠NPM = ∠NMP = a.
∠NPM = a
2
Form an equation by putting the sum of
the angles on one side and 180° on the
other side of the equals sign.
∠NMP + ∠NPM + ∠MNP = 180°
a + a + 40° = 180°
3
Simplify by collecting like terms.
4
Use inspection or backtracking to solve
for a.
×2
+ 40
a
2a
2a + 40°
70°
140°
180°
÷2
2a + 40° = 180°
2a = 180° − 40°
2a = 140°
140°
a = ----------2
a = 70°
– 40
Interior and exterior angles of a triangle
The angles inside a triangle are called interior angles. If any side of a triangle is
extended outwards, the angle formed is called an exterior angle. The exterior angle and
the interior angle adjacent to it are supplementary and therefore add up to 180°.
interior
angles
B
exterior
angles
Interior angle
Exterior angle
A
C
HACB + HBCD = 180º
MQ Vic 7 fig 11-55
D
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Maths Quest 7 for Victoria
WORKED Example 6
B
Find the value of the pronumerals in the diagram at right.
THINK
1
n
WRITE
∠BAC (angle p) together with its
adjacent exterior angle (∠DAB) add up
to 180°. Furthermore, ∠DAB = 125°.
So form an equation and solve for p.
2
The interior angles of LABC add up to
180°. Identify the values of the angles
and form an equation.
3
Simplify by adding 83° and 55° and
then solve for n.
125º p
A
D
83º
C
∠BAC = p; ∠DAB = 125°;
∠BAC + ∠DAB = 180°
So p + 125° = 180°.
p = 180° − 125°
p = 55°
∠BCA + ∠BAC + ∠ABC = 180°
∠BCA = 83° ∠BAC = p = 55°
∠ABC = n
So 83° + 55° + n = 180°.
n + 138° = 180°
n = 180° − 138°
n = 42°
remember
remember
1. The sum of the interior angles in any triangle is equal to 180°.
2. The angles at the base of an isosceles triangle are equal in size.
3. An exterior angle of a triangle, and an interior angle adjacent to it, are
supplementary (that is, add up to 180°).
11B
11.2
WORKED
Example
3
Angles in a triangle
1 Find the value of the pronumeral in each of the following triangles.
a
b
c
96º
g
Angle sum
of a triangle
t
30º
55º
25º
40º
x
68º
d
e
60º
f
33º
30º
54º
Triangles
k
60º
f
z
60º
Chapter 11 Polygons and polyhedra
433
2 Find the value of the pronumeral in each of the following right-angled triangles.
a
b
c
45º
40º
a
25º
d
b
WORKED
Example
4
3 Find the value of the pronumeral in each of the following triangles.
a
b
c
64º
c
52º
55º
e
n
WORKED
Example
5
4 Find the value of the pronumeral in each of the following triangles.
a
b
c
u
k
28º
48º
d
d
e
t
f
57º
f
32º
70º
5 Find the missing angle in each of the following diagrams.
a
b
b
70°
60°
p
p
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Maths Quest 7 for Victoria
c
d
p
k
62°
50°
100°
6 a
b
c
d
WORKED
Example
6
62°
An isosceles triangle has 2 angles of 55° each. Find the size of the third angle.
An isosceles triangle has 2 angles of 12° each. Find the size of the third angle.
Two angles of a triangle are 55° and 75° respectively. Find the third angle.
Two angles of a triangle are 48° and 68° respectively. Find the third angle.
7 Find the value of the pronumerals in each of the following diagrams.
a
b
n
b
158º
130º p
a
130º
60º
c
d
125º
50º
s
y
x
e
130º
t
f
b
26º
55º
34º
72º
n
g
h
m
m
120º
t
t
56º
8 a Use a ruler and a protractor to construct each of the following triangles.
i An isosceles triangle with a base of 4 cm and equal angles of 50° each.
ii An isosceles triangle with two sides which are 5 cm each and two equal angles
which are 45° each.
b On your diagrams label the size of each angle. Classify the triangles according to
the size of their angles.
11.1
9 Below are sets of 3 angles. For each set state whether or not it is possible to construct
a triangle with these angles. Give a reason for your answer.
a 40°, 40°, 100°
b 45°, 60°, 70°
c 45°, 55°, 85°
d 111°, 34.5°, 34.5°
10 Explain in your own words why it is impossible to construct a triangle with 2 obtuse angles.
Chapter 11 Polygons and polyhedra
435
1
1 Name the triangle shown at right based on
the length of the sides.
2 Name the triangle in question 1 based on the size of its angles.
3 A triangle has angles of 35° and 76°. Find the third angle.
4 One angle of a right-angled triangle is 37°. Find the third angle.
In questions 5 to 9, find the missing angle in each triangle.
5
6
72º
7
38º
48º
a
57º
j
h
8
9
t
64º
y
22º
118º
10 Find the missing angle shown in the photograph.
x
41°
436
Maths Quest 7 for Victoria
Types of quadrilaterals
Any 2-dimensional closed shape with 4 straight sides is called a quadrilateral. Quadmeans four, as in quadruplets (four babies), or quadriplegic (paralysed in all four
limbs). Lateral means sides, as in lateral movement (sideways movement) or lateral
thinking (thinking sideways, or around, a problem).
All quadrilaterals can be divided into 2 major groups: parallelograms and other
quadrilaterals.
Parallelograms are quadrilaterals with both pairs of opposite sides being parallel
to each other. Parallelograms include rectangles, squares and rhombuses
(diamonds).
The table below shows different parallelograms and their properties. Note that
parallel sides are marked with identical arrows.
Parallelogram
Shape
Properties
Parallelogram
Opposite sides are equal in
length.
Opposite angles are equal in size.
Rectangle
Opposite sides are equal in
length.
All angles are the same and
equal to 90°.
Rhombus
All sides are equal in length.
Opposite angles are equal in size.
Square
All sides are equal in length.
All angles are the same and
equal to 90°.
Quadrilaterals
Rectangles
Rhombuses
Squares
Other quadrilaterals include trapeziums, kites and irregular quadrilaterals. The
following table shows properties of these shapes.
Chapter 11 Polygons and polyhedra
Other
quadrilaterals
Shape
437
Properties
Trapezium
One pair of opposite sides is parallel.
Kite
Two pairs of adjacent (next to each other)
sides are equal in length.
One pair of opposite angles (the ones that
are between the sides of unequal
length) are equal.
Irregular
quadrilateral
This shape does not have any special
properties.
WORKED Example 7
Name the following quadrilaterals, giving reasons for your answers.
a
b
THINK
WRITE
a The markings on this quadrilateral
indicate that all sides are equal in length
and all angles equal 90°. Classify the
quadrilateral by finding the matching
description in the table.
a The given quadrilateral is a square, since all
sides are equal and all angles are 90°.
b The arrows on the sides of this
quadrilateral indicate that there are 2
pairs of parallel sides. Find the matching
description in the table and hence name
the quadrilateral.
b The given quadrilateral is a parallelogram,
since it has 2 pairs of parallel sides.
remember
remember
1. A quadrilateral is a 2-dimensional closed shape with 4 straight sides.
2. All quadrilaterals can be divided into 2 major groups: parallelograms and other
quadrilaterals.
3. Parallelograms have 2 pairs of parallel sides and include rectangles, squares
and rhombuses.
4. Other quadrilaterals include trapeziums, kites and irregular quadrilaterals.
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Maths Quest 7 for Victoria
11C
Types of quadrilaterals
1 Name the following quadrilaterals, giving reasons for your answers.
a
b
c
7
WORKED
Example
Quadrilaterals
Rectangles
d
e
f
Rhombuses
Squares
2 multiple choice
a
b
c
This quadrilateral is a:
A square
B rectangle
D rhombus
E parallelogram
This quadrilateral is a:
A trapezium B parallelogram
D irregular quadrilateral
This quadrilateral is a:
A trapezium B square
C irregular quadrilateral
E parallelogram
3 State whether each of the following statements is true or false.
a All squares are rectangles.
b All squares are rhombuses.
c All rectangles are squares.
d Any rhombus with at least one right angle is a square.
e A rectangle is a parallelogram with at least one angle equal to 90°.
f A trapezium with 2 adjacent right angles is a rectangle.
g All rhombuses are kites.
h A kite could be a parallelogram.
C kite
C rhombus
E kite
D kite
Chapter 11 Polygons and polyhedra
439
4 multiple choice
A rectangle is a quadrilateral because:
A it has 4 right angles
B it has 2 pairs of parallel sides
C its opposite sides are equal in length
D it has 4 straight sides
E it has 2 pairs of parallel sides and 4 right angles.
5 Draw 4 equilateral triangles with side lengths 4 cm and cut them out.
a Use 2 of these triangles to make a rhombus. Draw your solution.
b Use 3 of these triangles to make a trapezium. Draw your solution.
c Use all 4 triangles to make a parallelogram. Draw your solution.
6 Copy and cut out the following set of shapes.
Arrange the shapes to form a square.
Draw your solution.
7 State the types of quadrilaterals that can be seen in each of the following pictures.
a
b
8 In your house, find an example of each type of quadrilateral discussed in this section.
Write down the type of quadrilateral and where you found it.
9 The picture at right is made up of equilateral triangles.
How many rhombuses can you find in the picture?
(One rhombus that is made up of 2 triangles is shown.)
440
Maths Quest 7 for Victoria
Angles in a quadrilateral
Sum of angles in a quadrilateral
You will need a ruler and a protractor.
1. Draw an irregular quadrilateral in your workbook.
2. Measure each of the 4 angles using a protractor.
3. Find the sum of the 4 angles.
4. Record your results for the irregular quadrilateral into the table below.
Quadrilateral
First
angle
Second
angle
Third
angle
Fourth
angle
Sum of
angles
Irregular
quadrilateral
Trapeziums
Parallelogram
Trapezium
Kite
Parallelograms
Kites
Square
5. Repeat steps 1–4 for each of the other quadrilaterals in the table.
6. Study your results and write any patterns that you have noticed, regarding the
sum of angles in a quadrilateral.
As a result of your investigation you should have discovered the rule that is stated
below.
c
Angle sum in a
quadrilateral
The sum of the angles in any quadrilateral is 360°.
In the quadrilateral at right
a + b + c + d = 360°
b
a
d
This can be easily demonstrated.
C
B
x
y
In quadrilateral ABCD shown above,
t
the diagonal BD has been drawn. This
diagonal divides the quadrilateral into 2 triangles:
triangle ABD and triangle BCD.
z
s
u
In triangle ABD: ∠s + ∠t + ∠u = 180°.
A
D
In triangle BCD: ∠z + ∠x + ∠y = 180°.
So in both triangles together ∠s + ∠t + ∠u + ∠z + ∠x + ∠y = 180° + 180°
or
∠s + ∠t + ∠u + ∠x + ∠y + ∠z = 360°.
[1]
On the other hand, in the quadrilateral ABCD: ∠DAB = ∠s; ∠ABC = ∠t + ∠x;
∠BCD = ∠y and ∠CDA = ∠u + ∠z.
441
Chapter 11 Polygons and polyhedra
And so ∠DAB + ∠ABC + ∠BCD + ∠CDA = ∠s + ∠t + ∠x + ∠y + ∠u + ∠z.
That is, ∠s + ∠t + ∠u + ∠x + ∠y + ∠z = sum of angles in the quadrilateral.
[2]
Placing the two results next to each other, we have:
∠s + ∠t + ∠u + ∠x + ∠y + ∠z = 360°
∠s + ∠t + ∠u + ∠x + ∠y + ∠z = sum of angles in the quadrilateral.
[1]
[2]
Comparing [1] and [2] we observe that the left-hand sides of both equations are the
same. Therefore, the right-hand sides of the equations must also be equal and so the
sum of angles in a quadrilateral = 360°.
We can use this rule to find missing angles in quadrilaterals, as shown in the
examples that follow.
WORKED Example 8
Find the value of the pronumeral in the diagram at right.
120º
80º
75º
b
THINK
1
2
3
The sum of the angles in a quadrilateral
is 360°. So express this as an equation.
Simplify by adding 120°, 80° and 75°.
Solve to find the value of b.
WRITE
b + 80° + 75° + 120° = 360°
b + 275° = 360°
b = 360° − 275°
b = 85°
WORKED Example 9
Find the value of the pronumeral in the following
diagram, giving a reason for your answer.
x
72º
THINK
WRITE
According to the markings, the opposite
sides of the given quadrilateral are parallel
and equal in length. Therefore, this
quadrilateral is a parallelogram. In a
parallelogram opposite angles are equal. So
state the value of the pronumeral.
Opposite angles in a parallelogram are equal in
size. Therefore, x = 72°.
442
Maths Quest 7 for Victoria
WORKED Example 10
Find the value of the pronumerals in the following diagram.
THINK
1
2
3
4
5
50º
WRITE
t
136º
k + t + 50° + 136° = 360°
Form an equation by writing the sum of
the angles on one side and 360° on the
other side of an equals sign.
The quadrilateral shown in the diagram
is a kite. Angle t and angle 136° are the
angles between unequal sides and
therefore must be equal in size.
Replace t in the equation with 136°.
Simplify.
Solve to find the value of k.
k
t = 136°
k + 136° + 50° + 136° = 360°
k + 322° = 360°
k = 360° − 322°
k = 38°
remember
remember
The sum of angles in any quadrilateral is equal to 360°.
11D
WORKED
Example
8
Angle sum in a
quadrilateral
Angles in a quadrilateral
1 Find the value of the pronumeral in each of the following
diagrams.
a
b
c
t
42º
115º
110º
42º
138º
b
d
e
18º
Angles in a
quadrilateral
t
50º
f
54º
120º
107º
107º
t
m
20º
p
g
h
127º
250º
s
i
32º
c
k
110º
12º
93º
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Chapter 11 Polygons and polyhedra
WORKED
Example
9
2 Find the value of the pronumeral in each of the following diagrams, giving reasons for
your answers.
a
b
m
75º
78º
u
c
d
132º
108º
t
f
e
f
63º
73º
z
p
WORKED
Example
10
3 Find the value of the pronumerals in each of the following diagrams.
a
b
c
c
d
98º
m
t
m
c
t
82º
64º
36º
d
e
f
106º
p
96º
x
91º
n
115º
75º
p
m
m
t
4 multiple choice
The value of t in the following diagram is:
A 360°
B 112°
C 222°
D 138°
E 180°
t
42º
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Maths Quest 7 for Victoria
5 multiple choice
117º
The value of r in the following diagram is:
A 117°
B 63°
C 234°
D 126°
E 57°
r
6
This photograph shows the roof of a fast food restaurant.
Calculate the value of p.
119°
p
7 Find the size of the obtuse angle in the kite shown at
right.
8 Two angles in a parallelogram are 45° and 135°.
Find the other 2 angles.
65°
9 Tom measures 2 angles of a kite at 60° and 110°,
but forgets which angle is which. Draw 3 different
kites that Tom may have measured, showing the size of all angles in each diagram.
10 Below are sets of 4 angles. For each of the sets decide whether it is possible to construct a quadrilateral. Explain your answer.
a 25°, 95°, 140°, 100°
b 40°, 80°, 99°, 51°
GAM
me
E ti
Polygons and
polyhedra 01
11 Three angles of a quadrilateral are 60°, 70° and 100°.
a What is the size of the fourth angle of this quadrilateral?
b How many quadrilaterals with this set of angles are possible?
c Construct one quadrilateral with the given angle sizes in your book. (The choice of
the length of the sides is yours.)
Constructing quadrilaterals
1. (a) Is it possible to construct a quadrilateral with:
(i) 2 obtuse angles?
(ii) 3 obtuse angles?
(iii) 4 obtuse angles?
(b) Explain your answer in each case. If possible, construct one quadrilateral
of each type in your book.
(c) Based on your answers to part (a), complete the following sentence: ‘The
maximum possible number of obtuse angles in a quadrilateral is . . .’
2. Is it possible to construct a quadrilateral with 4 acute angles? If this is possible,
construct one such quadrilateral in your book. If this is not possible, explain
why it is so.
3. Construct quadrilaterals with exactly 2 right angles so that these right angles are:
(a) adjacent (that is, next to each other)
(b) opposite to each other.
Name the shapes that you have constructed.
4. Is it possible to construct a quadrilateral with:
(a) exactly 1 right angle?
(b) exactly 3 right angles?
Give reasons for your answers.
Chapter 11 Polygons and polyhedra
445
What did Doroth
Dorothyy’s 3 friends in
the Wizar
Wizard
d of Oz want from
from
the wizar
wizard?
d?
The size of the
55°
angles represented by letters in
each of the triangles gives the
puzzle answer code.
131°
64°
75°
E
A
100°
N
77°
40°
B
81°
125°
87°
O
I
A
87°
36°
D
A
48°
42°
64°
15°
C
103°
96°
59°
71°
N
U
61°
42°
37°
72°
39°
E
123°
28°
86°
96°
43°
94°
R
A
56°
R
78°
40°
58°
86°
G
T
41°
R
55°
27°
H
38°
84°
68°
82°
150°
80°
145°
78°
A
20°
A
66° 105° 43° 47° 18° 75° 40° 121° 162° 86° 90° 144° 97°
22°
35°
199°
15°
91°
130°
65°
60°
206°
MQ 7 Chapter 11 Page 446 Thursday, September 13, 2001 3:45 PM
446
Maths Quest 7 for Victoria
Design for a front gate
Val and Peter want to replace their front gate with another of the same design.
To have this gate made, they need to supply a diagram of it with all measurements
and angles shown.
8 cm
27°
18 cm
60 cm
1m
1. There are 4 different shapes formed by the metal bars of the gate. How many
different types of triangles are there? Can you name them?
2. How many types of quadrilaterals are there? Name them.
3. Draw a diagram of the gate showing the length measurements and the one angle
that is given.
4. Use this angle to calculate all the remaining angles in the diagram.
5. Explain how you were able to achieve this.
Your turn!
Using a ruler and protractor, design a fence that is to be constructed using metal
bars. Include different triangles and quadrilaterals to make your design as
interesting as possible. Write a short report describing the shapes you have used
and important angles which need to be marked on your design to assist in the
construction of the fence.
Chapter 11 Polygons and polyhedra
447
Polygons
A polygon is any closed shape with 3 or more sides, each of which is a straight line.
WORKED Example 11
Which of the following shapes are polygons?
a
b
c
THINK
WRITE
a The shown shape is closed and all of its
sides are straight lines. So by definition
this shape is a polygon.
a The shape is a polygon.
b Although all sides of this shape are
straight lines, it is not closed and hence
is not a polygon.
b The shape is not a polygon.
c The shape is closed, but one of the sides
is not straight. Therefore this shape is
not a polygon.
c The shape is not a polygon.
Naming polygons
Polygons are named according to the number of sides or angles in the shape. (Note that
the number of sides in any polygon is equal to the number of angles in the polygon.)
The table below gives the names of the most common polygons.
Number of sides
Name
Number of sides
Name
3
triangle
9
nonagon
4
quadrilateral
10
decagon
5
pentagon
11
undecagon
6
hexagon
12
dodecagon
7
heptagon
20
icosagon
8
octagon
448
Maths Quest 7 for Victoria
Can you see why this famous building in the USA is called the Pentagon?
WORKED Example 12
Name the following polygons.
a
b
c
THINK
WRITE
a
a Number of sides = 5
1
2
Count the number of sides in the
polygon.
Match the number of sides with the
corresponding name in the table.
The polygon is a pentagon.
b Repeat steps 1 and 2 as in part a.
b Number of sides = 10
The polygon is a decagon.
c Repeat steps 1 and 2 as in part a.
c Number of sides = 8
The polygon is an octagon.
remember
remember
1. A polygon is a closed shape with straight sides.
2. Polygons are named according to the number of sides or angles in the shape.
Chapter 11 Polygons and polyhedra
11E
WORKED
Example
11
449
Polygons
1 Which of the following shapes are polygons?
a
b
Polygons
Regular
polygons
c
WORKED
Example
12
d
2 Name the following polygons.
a
b
c
d
e
f
3 Draw 2 different examples of each of the following polygons.
a hexagon
b quadrilateral
c
nonagon
d pentagon
e octagon
f
triangle
450
Maths Quest 7 for Victoria
4 Count the number of sides on a 50 cent piece and
name its shape.
5 multiple choice
What shape is each of the following?
a
A quadrilateral
C octagon
E heptagon
b A horizontal cross-section of a pencil
A quadrilateral
B hexagon
D pentagon
E circle
B hexagon
D pentagon
C octagon
6 These patterns are made up of different polygons. Can you name them all?
a
b
c
d
11.2
7 Name one place where you can find these polygons in your home or school.
a a triangle
b a quadrilateral
c a hexagon
d an octagon
Chapter 11 Polygons and polyhedra
451
Constructing polygons
Trace this shape, cut it out and cut along the dotted lines
to make 3 pieces. Rearrange the pieces to make the
following polygons. (Draw the solutions in your workbook).
(a) triangle
(b) square
(c) rectangle
(d) trapezium
(e) pentagon
(f) parallelogram
2
1 Name the quadrilateral shown, giving a reason for
your answer.
For questions 2 to 5, find the missing angle in each of the quadrilaterals.
2
94º
a
4
3
117º
h
61º
5
71º
109º
77º
65º
w
t
39º
6 A quadrilateral has 3 angles measuring 56°, 102° and 79°. Find the size of the fourth
angle.
7 A rhombus has two angles each of size 78°. Find the size of each of the other two
angles.
8 Name the polygon at right.
9 Draw two different examples of an octagon.
10 What name is given to a polygon with 11 sides?
452
Maths Quest 7 for Victoria
A. van Leeuwenhoek
Leeuwenhoek of Holland was
the first person to do this in 16
1676.
7 TRAPEZIUM
Match up the letter in each polygon
with the number beside the correct name
for the shape to find the answer code.
O
E
8 EQUILATERAL
TRIANGLE
1 PARALLELOGRAM
9 OBTUSE-ANGLED,
SCALENE TRIANGLE
2 RIGHT-ANGLED,
SCALENE TRIANGLE
10 RHOMBUS
3 ACUTE-ANGLED,
ISOSCELES TRIANGLE
11 REGULAR PENTAGON
4 SQUARE
12 OBTUSE-ANGLED,
ISOSCELES TRIANGLE
5 REGULAR HEXAGON
13 RECTANGLE
6 ACUTE-ANGLED,
SCALENE TRIANGLE
14 RIGHT-ANGLED,
ISOSCELES TRIANGLE
D
U
N
C
G
H
P
T
S
I
1
13
2
3
4
4
11
M
5
10
6
7
4
R
8 6
5
9
9 10 11 12 11
2
11
5
2
4
14
1
8
6
Chapter 11 Polygons and polyhedra
453
Constructing polygons
In the preceding exercises there were some questions involving construction of triangles and quadrilaterals. In this section we will discuss further ways of constructing
polygons using a ruler, compass and protractor.
A polygon with all sides equal in length and all angles equal in size is called a
regular polygon.
A regular polygon can be constructed in a circle as shown in the following worked
example.
WORKED Example 13
Construct a regular nonagon in a circle of radius 5 cm.
THINK
1
2
3
4
WRITE/DRAW
A nonagon has 9 vertices, so we need
to mark 9 points on the circumference.
Furthermore, since the nonagon is
regular, the vertices must be equidistant
from each other (that is, evenly spaced
along the circumference). There are
360° in a circle, so divide 360° by 9, to
find the distance between each point on
the circumference.
Draw a circle of radius 5 cm.
Use a protractor to mark off 9 points on
the circle at 40° intervals. (These points
are to become the vertices of the
nonagon.)
Number of sides = 9, so there are
360° ÷ 9 = 40° between each point.
90
40
0
Join the points with straight lines to
construct a nonagon.
remember
remember
1. A regular polygon has all sides of equal length and all angles of equal size.
2. To construct a regular polygon in a circle, first divide 360° by the number of
sides. This gives you the angle between the vertices. Then mark off the points
on the circumference of the circle and join them together with straight lines.
454
Maths Quest 7 for Victoria
11F
11.3
WORKED
Example
Constructing polygons
1 Construct a square in a circle of radius 5 cm.
13
2 Construct a regular pentagon in a circle of radius 5 cm.
3 Construct a regular hexagon in a circle of radius 5 cm.
4 Construct a regular octagon in a circle of radius 5 cm.
Internal
angle of a
polygon
5 a Follow the instructions below to produce this design.
i Draw a square of side 10 cm.
ii Find the midpoint (middle) of each side.
iii Join the midpoints to form a new square.
iv Repeat steps ii and iii for your new square.
b Repeat steps ii and iii to produce smaller and smaller
squares. How many smaller squares can you make?
c Colour in your final design.
6 Draw your solutions for each part of this question in your book.
First draw a square with sides of length 5 cm.
a With 1 straight line divide the square into:
i 2 equal rectangles
ii 2 equal right-angled triangles
iii 2 equal trapeziums.
b With 2 straight lines divide the square into:
i 4 equal triangles
ii 4 equal squares
iii 3 equal rectangles
iv 4 equal irregular quadrilaterals.
c With 3 straight lines divide the square into:
i 4 equal rectangles
ii 6 equal rectangles.
7 Copy the following shape and cut it out. Cut out along the
dotted lines and rearrange the pieces to form a square.
(Draw the solution in your workbook.)
8 multiple choice
GAM
me
E ti
Polygons and
polyhedra 02
Which of the following is a regular quadrilateral?
A A rectangle
B A parallelogram
D A square
E All of the above
C A rhombus
Chapter 11 Polygons and polyhedra
455
Plans and views
An object can be viewed from different angles. Architects and draftspersons often draw
plans of building sites and various objects when viewed from the front, the side or the
top.
The front view, or front elevation, is what you see if you are standing directly in front
of an object.
The side view, or side elevation, is what you see if you are standing directly to one
side of the object. You can draw the left view or the right view of an object.
The top view, or bird’s eye view, is what you see if you are hovering directly over the
top of the object looking down on it.
WORKED Example 14
The following object is made from 4 cubes.
Draw plans of it showing:
a the front view
b the right view
c the top view.
THINK
DRAW
a Make this shape using cubes. Place the
shape at a considerable distance and
look at it from the front (this way you
can see only the front face of each cube).
Draw what you see. (Or simply imagine
looking at the shape from the front and
draw what you see.)
a
b Look at your model from the right, or
imagine that you can see only the right
face of each cube and draw what you
see.
b
c Look at your model from the top, or
imagine that you can see only the top
face of each cube. Draw what you see.
c
Front
Front view
Right view
Top view
Front
456
Maths Quest 7 for Victoria
WORKED Example 15
Draw:
i the front view
ii the right view
iii the top view of this solid.
THINK
1
2
3
DRAW
Find an object of similar shape, or
visualise the object in your head.
Whether viewed from the front, or from
the right of the object, the cylindrical
shaft will appear as a long thin
rectangle. The circular discs will also
be seen as a pair of identical rectangles.
So the front view and the right view are
the same.
When the object is viewed from above,
all we can see is the flat surface of the
top disc; that is, a large and a small
circle with the same centre. (Note that
such circles are called concentric.)
Front view
Right view
Top view
WORKED Example 16
The front, right and top views of a solid are shown.
Use cubes to construct the solid.
Front
Front
view
THINK
1
2
CONSTRUCT
Use cubes to construct the solid.
Check carefully that your solid matches
each of the 3 views you are given.
Make adjustments if necessary.
Front
Right
view
Top
view
Chapter 11 Polygons and polyhedra
457
remember
remember
1. The front view, or front elevation, is what you see if you are standing directly in
front of an object.
2. The side view, or side elevation, is what you see if you are standing directly to
one side of the object. You can draw the left view or the right view of an object.
3. The top view, or bird’s eye view, is what you see if you are hovering directly
over the top of the object looking down on it.
11G
Plans and views
1 The following objects are made from cubes. For every object draw the plans, showing
the front view, the right view and the top view. (You may wish to use a set of cubes or
14
building blocks to help you.)
a
b
c
WORKED
Example
Front
Front
Front
d
e
f
Front
Front
Front
g
h
Front
Front
458
Maths Quest 7 for Victoria
i
j
Front
Front
WORKED
Example
15
2 Draw the front, right and top views of each solid shown.
a
b
c
d
3 The front, right and top views of a solid are shown. In each case, use cubes to construct
the solid.
16
a
b
WORKED
Example
Front view
Front view
Right view
Right view
Top view
Top view
Front
Front
c
d
Front view
Front view
Right view
Right view
Top view
Top view
Front
Front
Chapter 11 Polygons and polyhedra
459
4 multiple choice
The front, right and top views of a solid are shown. Which of the given drawings could
represent the solid?
Front view
Right view
Top view
Front
A
B
C
Front
Front
D
Front
E
Front
Front
5 a What shape is the top view of a telephone pole?
b What shape is the top view of the
Melbourne Cricket Ground?
c What shape is the side view of a bucket?
d What shape is the top view of a car?
6 a Draw the side view of a pool table.
b Draw the front view of your house
(seen from the street).
c Draw the side view of a kettle.
d Draw the top view of your television set.
7 A shape is made using only 4 cubes.
Its front view, right view and top
view are shown.
Front view
Right view
Top view
Front
a Is it possible to construct this solid?
b Describe or draw what this solid would look like.
460
Maths Quest 7 for Victoria
Polyhedra, nets and Euler’s rule
A polyhedron is a 3-dimensional shape in which each flat surface is a polygon.
This photograph of an ancient Egyptian pyramid is an example of a polyhedron, since
each of its 5 flat surfaces is a polygon.
The flat surfaces which make a polyhedron are
called faces.
The lines where 2 faces of a polyhedron meet are
called edges.
The points where 3 or more edges of a polyhedron
meet are called vertices.
Edge
Vertex
Face
WORKED Example 17
For the polyhedron shown, write down:
a the number of faces and the shape of each face
b the number of edges
c the number of vertices.
THINK
WRITE
a The base of the polyhedron shown is a
square and the other 4 faces are
triangles.
a Number of faces = 5
Shape of the faces: 1 square and 4 triangles
b Four edges are formed where each of the
triangles meets the square base. Another
4 edges are formed where the triangular
faces meet each other.
b Number of edges = 8
c There are 4 vertices on the square base
and 1 at the top.
c Number of vertices = 5
Chapter 11 Polygons and polyhedra
461
Naming polyhedra
The table below gives the names of some common polyhedra that you might often see.
The names are often associated with the number of faces of the polyhedron.
Polyhedron
Number of
faces
Name
4
tetrahedron, or triangular-based
pyramid
5
square-based pyramid
6
cube
8
octahedron
12
dodecahedron
20
icosahedron
462
Maths Quest 7 for Victoria
Poly
Technology and polyhedra
A demonstration version of the program Poly is available on the Maths Quest 7
CD-ROM. This program allows you to visualise polyhedra and their nets.
When you first open Poly, follow these steps to select the most appropriate options:
Go to View then select Available modes. ‘Tick’ the following options:
Option 2:
3-dimensional shaded polyhedra
Option 4:
3-dimensional edges (wireframe)
Option 5:
3-dimensional vertices
Option 6:
2-dimensional net
Poly can be used to assist you in counting the number of faces, edges and vertices as
well as view the shape of each face.
For the polyhedron in worked example 17, follow these steps:
1. Select Johnson Solids and Square Pyramid (J1)
2. Press the icon
(3-dimensional shaded polyhedra) and rotate the object to
count the number of faces and to see the shape of each face. (You can rotate the solid
by placing your mouse arrow over the solid then clicking and holding down the
mouse while moving the arrow.)
3. Press the icon
(wireframe). Rotate until all edges are clearly seen and can be
counted.
4. Press the icon
(3-dimensional vertices). Rotate until all vertices are clearly
seen and can be counted.
Use Poly to obtain different views of the polyhedra shown in the table by rotating
each solid using option
(3-dimensional shaded polyhedra).
Chapter 11 Polygons and polyhedra
463
Nets of polyhedra
A 2-dimensional plan that can be folded to construct a 3-dimensional polyhedron
is called a net of that polyhedron.
This is a net of a square-based pyramid:
Square-based pyramid
if you fold up the triangles and stick them together,
the square-based pyramid shown at right will be formed.
(2-dimensional
net). You can also see how the solid ‘unfolds’ into a net and then folds back into a polyhedron by moving the button forward and backwards along the horizontal slot when
using option
Poly
Use Poly to see the nets of different solids by selecting option
(3-dimensional shaded polyhedra).
remember
remember
A polyhedron is a 3-dimensional shape in which every flat surface is a polygon.
The flat surfaces that form a polyhedron are called faces.
The lines where 2 faces of the polyhedron meet are called edges.
The points where 3 or more edges of the polyhedron meet are called vertices.
The net is a 2-dimensional plan, which can be folded to form a 3-dimensional
object.
11H
WORKED
Example
Polyhedra, nets and Euler’s
rule
You may wish to use the program Poly to assist you in completing this exercise.
1 i
ii
iii
17
For each polyhedron shown, write down:
a the number of faces and the shape of each face
b the number of edges
c the number of vertices.
Poly
1.
2.
3.
4.
5.
464
Maths Quest 7 for Victoria
2 Name each of the polyhedra in question 1.
3 Copy this net of a tetrahedron. Tabs have been included
to assist in the construction.
Cut out the net and fold it to construct the tetrahedron.
Look at the tetrahedron and write down:
a the number of faces
b the number of vertices
c the number of edges.
4 This is the net of a parallelepiped.
Each of its faces is a parallelogram.
Copy the net, cut it out and
fold it to construct the
parallelepiped. Look at the
parallelepiped and write down:
a the number of faces
b the number of vertices
c the number of edges.
5 This is the net of a truncated
tetrahedron. It is a tetrahedron
with its corners cut off.
Copy the net, cut it out and
fold it to construct the
truncated tetrahedron. Look
at the truncated tetrahedron
and write down:
a the number of faces
b the number of vertices
c the number of edges.
6 Copy this net of an icosahedron.
Cut out the net and fold it to construct the icosahedron. Look at the icosahedron and
write down:
a the number of faces
b the number of vertices
c the number of edges.
MQ 7 Chapter 11 Page 465 Wednesday, June 18, 2003 4:21 PM
Chapter 11 Polygons and polyhedra
465
7 a Copy this table into your workbook. Use the answers to questions 1, 3, 4, 5, 6 (or
the program Poly) to complete the table.
Question
number
Name of polyhedron
1i
cube
1 ii
octahedron
1 iii
dodecahedron
3
tetrahedron
4
parallelepiped
5
truncated tetrahedron
6
icosahedron
Number
of faces
(F)
Number
of vertices
(V)
Number
of edges
(E)
F+V−E
b State the pattern that you have found.
c
The pattern that you have discovered is known as Euler’s rule. (Euler is pronounced ‘Oiler’.) Copy the rule into your workbook for future reference.
Euler’s rule for polyhedra can be stated as:
for any polyhedron, F + V − E = 2,
where F is the number of faces, V is the number of vertices and E is the number
of edges of the polyhedron.
In other words, the number of faces plus the number of vertices minus the number of
edges equals 2.
8 Use Euler’s rule, established in the previous question, to check whether it is possible
for a polyhedron to have:
a 6 faces, 8 vertices and 10 edges
b 9 faces, 12 vertices and 19 edges.
9 Use Euler’s rule to answer each of the following questions.
a A polyhedron has 10 faces and 16 vertices. How many edges does it have?
b A polyhedron has 12 faces and 18 vertices. How many edges does it have?
c
A polyhedron has 10 edges and 6 vertices. How many faces does it have?
d A polyhedron has 8 faces and 12 vertices. How many edges does it have?
Using
Euler’s
rule
466
Maths Quest 7 for Victoria
10 a Use straws and plasticine (or any other suitable material) to construct these polyhedra:
i a square-based pyramid
ii a nonahedron
11.3
QUEST
GE
S
EN
MAT H
b For each model count the number of faces, edges and vertices and hence verify
Euler’s rule.
CH
AL
1 How many rectangles can you find in this shape?
L
2 Show how to cut this rectangle into 2 pieces that fit together to form a
square?
Poly
Making models of polyhedra
Use the program Poly to find the nets of 3 polyhedra. There are some suggestions
below. Print out each net and trace onto coloured paper or card. (You may like to
enlarge your net by using a photocopier first.) Cut and fold to form each of these
polyhedra. Have fun!
• Hebesphenomegacorona (J89) (in Johnson Solids)
• Decagonal deltohedron (in Dipyramids and Deltohedrons)
• Hexakis octahedron (in Catalan Solids)
• Square anti-prism (in Prisms and Anti-prisms)
• Great rhombicosidodecahedron (in Archimedean Solids)
Chapter 11 Polygons and polyhedra
467
summary
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
Copy the sentences below. Fill in the gaps by choosing the correct word or
expression from the word list that follows.
An obtuse-angled triangle has one angle
90°.
An
triangle has two sides equal in length and two angles
equal in size.
An equilateral triangle has all sides
in length and all angles
equal in size.
An
triangle has all angles less than 90°.
A
triangle has no equal sides or angles.
A right-angled triangle has one angle equal to 90° (that is, a
angle).
A kite has two pairs of
sides, equal in length. The angles
between
sides of a kite are equal in size.
A
has two pairs of opposite sides equal in length and four 90° angles.
A square has four
and four 90° angles.
A
has one pair of parallel sides.
A rhombus has four equal sides. Opposite angles of a rhombus are
.
A parallelogram has two pairs of
sides. Opposite sides of a
parallelogram are equal in length and
angles are equal in size.
A polygon is any
2-dimensional shape with
edges.
A
polygon has all sides equal in length and all angles equal in size.
A polyhedron is a
object. Each face of a polyhedron is a
.
The sum of the angles in any triangle is equal to
degrees.
An
angle of a triangle and an interior angle adjacent to it are
.
The sum of the angles in any quadrilateral is equal to
degrees.
A plan which can be cut out and folded to make a
is called
a
of that shape.
The bird’s eye view of an object is its
view.
The
view is drawn when standing directly in front of an object.
The
is the left, or the right view of an object.
Euler’s rule for a polyhedron states that the number of
plus the
number of
minus the number of
equals 2.
WORD
faces
adjacent
polygon
closed
net
supplementary
acute-angled
rectangle
LIST
right
opposite
vertices
polyhedron
greater than
180
straight
top
edges
side elevation
unequal
equal sides
equal in size
equal
360
3-dimensional
front
regular
exterior
isosceles
scalene
parallel
trapezium
468
Maths Quest 7 for Victoria
CHAPTER
review
11A
1 Name the following triangles according to the length of their sides.
a
b
c
11A
2 Name the following triangles according to the size of their angles.
a
b
c
11B
3 Find the value of the pronumeral in each of the following.
a
b
t
c
62º
40º
b
48º
65º
x
11B
4 Find the value of the pronumeral in each of the following.
a
b
m
c
65º
n
p
52º
n
n
62º
11B
5 The Indian teepee shown has an angle of 46° at its peak. What angle
does the wall make with the floor?
46º
w
Chapter 11 Polygons and polyhedra
6 Name the following quadrilaterals, giving reasons for your answers.
a
b
c
d
e
85º
120º
x
11C
f
7 Find the value of the pronumeral in each of the following.
a
b
110º
11D
c
36º
240º
105º
80º
469
n
42º
e
8 Find the value of the pronumeral in each of the following.
a
b
g
g
11D
c
110º
k
w
68º
9 A circus trapeze attached to a rope is shown.
Find the size of angle t.
65º
65º
t
b
11 Draw two different examples of a hexagon.
11D
t
10 Name the following polygons.
a
126°
74º
c
11E
11E
470
Maths Quest 7 for Victoria
11F
12 Draw a circle of radius 5 cm. Mark points on its circumference every 120°. Join the dots
together to construct a triangle.
11F
13 Draw a circle of radius 5 cm.
a To construct a hexagon you need to mark 6 points on the circle’s circumference.
Calculate the number of degrees between each point.
b Mark off 6 points on the circle and construct a hexagon.
11G
14 Draw the front, side and top views of each of these solids.
a
b
Front
Front
11G
15 The front, side and top view of a solid are shown. Construct this solid, using blocks.
a
b
Front view
Front view
Right view
Right view
Top view
Front
Top view
Front
11H
16 For each polyhedron shown write down:
i the number of faces
ii the number of vertices
iii the number of edges.
a
b
17 a A polyhedron has 10 faces and 8 vertices. How many edges does it have?
b A polyhedron has 6 faces and 12 edges. How many vertices does it have?
c Is it possible to have a polyhedron with 10 faces, 6 edges and 4 vertices? Give reasons
test
for your answer.
yourself
CHAPTER
11H
11