CHAPTER
1
Symmetry and
Surface Area
GET READY
2
Math Link
4
1.1 Warm Up
5
1.1 Line Symmetry
6
1.2 Warm Up
15
1.2 Rotation Symmetry and Transformations
16
1.3 Warm Up
25
1.3 Surface Area
26
Graphic Organizer
36
Chapter 1 Review
37
Key Word Builder
40
Chapter 1 Practice Test
41
Math Link: Wrap It Up!
43
Challenge
44
Answers
46
Name: _____________________________________________________
Date: ______________
Get Ready
Using Translations
Transformations include translations, reflections, and rotations.
A translation is a slide along a straight line.
It can be horizontal (↔), vertical (↕), or oblique (slanted 4 or 1 ).
y
A
This is a translation
3 units horizontally
to the right, R3.
A’
2
B
C
E
C’
This is a translation 3 units
horizontally to the right
and 2 units vertically down,
R3 and D2.
y
D'
F
x
2 x
0
-2
1.
B’
D
A transformation moves a
shape to a different position.
F'
E'
Describe each translation.
a)
B'
b)
y
A'
y
units
E
G
F
H
C'
up
x
x
B
A
E'
G'
F'
H'
C
Drawing Reflections
A reflection is a mirror image in a line of reflection.
A point and its reflection are the same perpendicular ( ⊥ ) distance
from the line of reflection.
2
N
line of reflection
N’
The line of reflection is a horizontal line at y = 1.
Both N and N′ are 1 unit from the line of reflection.
2.
y
0
2
Draw the reflection image for each figure.
a)
b)
y
y
r
line of
reflection
line of reflection
r
x
2
MHR ● Chapter 1: Symmetry and Surface Area
x
4 x
Name: _____________________________________________________
Date: ______________
Drawing Rotations
A rotation is a turn about a point or centre of rotation.
It can be clockwise (
) or counter-clockwise (
).
The centre of rotation is at A.
The rotation is 180° counter-clockwise (
) about A.
y
2
centre of
rotation
0 A
-2
2
x
-2
180° is halfway around a circle.
3.
180°
a) Rotate figure DEFG 90° clockwise about its centre of rotation, F.
Draw the rotation image. Label it D′E′F′G′.
D
y
G
4
b) Write the coordinates of D′, E′, F′, and G′.
D′ = (2,
)
E
E′ =
F
2
-2
F′ =
2
x
G′ =
Using Surface Area
Surface area is the sum of the areas of all the faces of a 3-D object.
Find the surface area of the right rectangular prism.
Front or back
A=l×w
=6×8
= 48 cm2
Area of front and back:
A = 2 × 48
= 96 cm2
Total surface area = 96 + 24 + 32
= 152 cm2
4.
Top or bottom
A=l×w
=6×2
= 12 cm2
Area of top and bottom:
A = 2 × 12
= 24 cm2
Ends
A=l×w
=2×8
= 16 cm2
Area of 2 ends:
A = 2 × 16
= 32 cm2
8 cm
6 cm
2 cm
Calculate the surface area of the right rectangular prism.
Front or back
Top or bottom
Ends
Get Ready ● MHR 3
Name: _____________________________________________________
Date: ______________
Math Link
Reflections on Our World
symmetry
● an image or object has symmetry if a line can divide it into 2 identical parts
● an image has symmetry if it can fit onto itself by reflection or rotation
1.
a) This flower has several lines of symmetry.
Place a Mira™ or mirror from the top left corner to the
bottom right corner.
Does this show a line of reflection? Circle YES or NO.
b) Use the Mira™ or a mirror to draw other lines of reflections.
How many can you draw?
2.
a) Use a ruler to draw a vertical line (↕) through the centre of the
dragonfly.
Is this a line of reflection? Circle YES or NO.
b) Are there any other lines of reflection? Circle YES or NO.
3.
The dashed line shows the line of reflection, r.
C
a) Use a Mira™ to help you draw the reflected image.
b) In the original image, the highest point is
c) In the new image, the lowest point is
D
E
r
.
.
d) The new image has been reflected
in the line of reflection.
(horizontally or vertically)
4.
a) Figure ABC has been translated to make figure A′B′C′.
Fill in the blanks to describe a rule for this translation.
y
5
spaces up.
4
B to B′: __________________________________________
2
A to A′: move
spaces right and
0
C’
A
3
1
C to C′: __________________________________________
A’
6
C
1
B
2
3
4
5
b) Describe a translation that would place the image for ABC in a different quadrant.
_______________________________________________________
IV I
_______________________________________________________
III
_______________________________________________________
4
MHR ● Chapter 1: Symmetry and Surface Area
B’
II
6
7
8
x
Name: _____________________________________________________
Date: ______________
1.1 Warm Up
1.
Line A is the line of reflection. Draw the reflection image of each figure.
Name the coordinates of the reflection image.
a)
y
4
B
A
2
b)
C
E
y
D
2
F
D
ℓ
2
0
ℓ
0
O
G
2 x
4 x
2
–2
–2
A′ (
,
)
D′
D′
B′ (
,
)
E′
O′
F′
G′
C′
2.
Line segments are
perpendicular ( ⊥ ) if they
meet at right angles.
Circle the diagrams that show 2 perpendicular line segments.
a)
C
A
J
b)
B
K
D
c)
E
L
I
G
d)
F
O
M
H
N
P
3.
Use a ruler to draw a perpendicular line segment.
1
20
2
30
3
40
4
50
5
60
6
70
7
80
8
90
9
100
10
4.
10
E
0
4 cm
0
a)
H
b)
3 cm
G
F
List 2 examples of perpendicular line segments in your classroom.
________________________________________________________________________________
________________________________________________________________________________
1.1 Warm Up ● MHR 5
Name: _____________________________________________________
Date: ______________
1.1 Line Symmetry
line of symmetry
a line that divides an object or design in half
● each side of the line is a mirror image of the other
● can be vertical (↕), horizontal (↔), or oblique (slanted 1, 4 )
● also called line of reflection
● a figure may have more than 1 line of symmetry
●
vertical line
of symmetry
horizontal line
of symmetry
oblique line
of symmetry
line symmetry
a type of symmetry where a line of symmetry divides an image or object into 2 identical,
reflected halves
● identical halves can be reflected in a vertical (↕), horizontal (↔), or oblique
(slanted 1, 4 ) line of symmetry
●
Link the Ideas
Working Example 1: Find Lines of Symmetry
Each diagram shows line symmetry.
For each part, use a different method to find the line(s) of symmetry.
State the number of lines of symmetry and describe each one.
a)
Solution
Use a Mira™.
There is
horizontal line of symmetry.
0
number of
lines of symmetry
6
description of line
MHR ● Chapter 1: Symmetry and Surface Area
Name: _____________________________________________________
Date: ______________
b)
vertical line
of symmetry
Solution
Count squares on the grid to find the lines of symmetry.
The same number of squares are above and below the
horizontal line of symmetry.
The same number of squares are to the left and right of
the vertical line of symmetry.
horizontal line
of symmetry
Number of lines of symmetry:
Description: One line is
is
(↔), and one line
(↕).
c)
Solution
Use folding. If the shape on each side of the fold is the same, the fold is a line of symmetry.
You can sketch the complete
figures to prove that the fold
lines are lines of symmetry.
You can fold this figure along 4 different lines to make mirror images:
There is 1
of symmetry, and 2
(↔) line of symmetry, 1
(↕) line
( 1, 4 ) lines of symmetry.
1.1 Line Symmetry ● MHR 7
Name: _____________________________________________________
Date: ______________
How many lines of symmetry does each figure have?
Describe each line of symmetry as vertical, horizontal, or oblique.
a)
Number of lines of symmetry:
Description: _________________________________________________________________
b)
Number of lines of symmetry:
Description: _________________________________________________________________
c)
Number of lines of symmetry:
Description: _________________________________________________________________
____________________________________________________________________________
d)
Number of lines of symmetry:
Description: _________________________________________________________________
8
MHR ● Chapter 1: Symmetry and Surface Area
Name: _____________________________________________________
Date: ______________
Working Example 2: Complete Drawings Using Symmetry
Each drawing shows half of a figure.
The dashed grey line shows a line of symmetry.
Draw the whole image for each figure.
a)
Solution
Method 1: Use Paper Folding
‰ Fold a piece of paper in half.
‰ Draw the figure so the line of symmetry is along the folded edge.
‰ Cut out the figure.
‰ Unfold the paper to show the whole figure.
Method 2: Use Measurement or Counting
B
B
C
B
C
C
X
A
D
A
D
B’
C’
D
A
B’
C’
‰ Draw the half figure on the grid. Label the vertices A, B, C, and D.
Mark the line of symmetry.
‰ Reflect the points that are not on the line of symmetry to the opposite side
of the line.
These are points B and
.
‰ Draw the reflected points the same distance from the fold line.
This distance must be perpendicular ( ⊥ ) to the fold line.
Literacy Link
Use B′ and C′ to
label the new positions
of B and C after a
transformation.
B′ is read as “B prime.”
So, BX = B′ X′ and CD =
.
‰ Join A to B′, B′ to C′, and C′ to D to draw the figure.
1.1 Line Symmetry ● MHR 9
Name: _____________________________________________________
Date: ______________
b)
Solution
‰ From each vertex, mark the perpendicular distance
on the opposite side of the line of symmetry.
‰ Connect the lines to draw the figure.
‰ Complete the figure.
C
B
B’
A
A’
F
D
‰ The whole figure is the letter H.
The line of symmetry is not part of the final figure.
Complete each shape. Use the line of symmetry.
a)
10
b)
MHR ● Chapter 1: Symmetry and Surface Area
E
C’
Name: _____________________________________________________
Date: ______________
Check Your Understanding
Communicate the Ideas
1.
Any rectangle has only 2 lines of symmetry.
Do you agree? Circle YES or NO.
Give 1 reason for your answer.
_______________________________________________________________________________
_______________________________________________________________________________
2.
Sasha says a parallelogram has 2 lines of symmetry.
Basil says it has 4 lines of symmetry.
Kendra says it has no lines of symmetry.
Who is correct? Circle SASHA or BASIL or KENDRA.
Give 1 reason for your answer. _______________________________________________________
________________________________________________________________________________
Practise
3.
Draw all the lines of symmetry on each figure.
a)
b)
c)
d)
1.1 Line Symmetry ● MHR 11
Name: _____________________________________________________
4.
Draw all the lines of symmetry on each diagram.
a)
b)
5.
Circle the figures that have only 2 lines of symmetry.
6.
The dashed line is the line of symmetry. Complete each diagram.
12
a)
b)
c)
d)
MHR ● Chapter 1: Symmetry and Surface Area
Date: ______________
Name: _____________________________________________________
Date: ______________
Apply
7.
a) The y-axis is the line of reflection.
Draw the reflection image.
Label the reflected vertices A′, C′, and E′.
y
A
5
E
b) What are the coordinates of the reflected image?
A′ (
,
C
)
-5
C′
0
5 x
E′
c) Do the original figure and the reflection image show line symmetry? Circle YES or NO.
Give 1 reason for your answer.
____________________________________________________________________________
8.
9.
How many lines of symmetry does each flag have?
a)
b)
c)
d)
Look at the upper-case letters of the English alphabet.
a) Which letters have a horizontal line of symmetry?
b) Which letters have a vertical line of symmetry?
c) Which letters have both horizontal and vertical lines of symmetry?
1.1 Line Symmetry ● MHR 13
Name: _____________________________________________________
Date: ______________
10. Using upper-case letters, you can write the word MOM vertically or horizontally.
The word has 1 vertical line of symmetry in each position.
Write another word that has 1 vertical line of symmetry when you print it
vertically or horizontally.
11. Write a 2-digit number that has 2 lines of symmetry. Show the lines of symmetry.
Math Link
Imagine you work for a company that designs playing cards.
Your job is to make an interesting design for a deck of cards.
Your design must have at least 1 line of symmetry.
Draw your design in the box or on the grid.
14
MHR ● Chapter 1: Symmetry and Surface Area
M
O
M
MOM
Name: _____________________________________________________
Date: ______________
1.2 Warm Up
1.
a) Translate the figure 4 units to the right.
Label the new coordinates A′, B′, and C′.
y
A
8
b) What are the coordinates of A′, B′, and C′?
A′ (
,
6
4
)
B
B′
C
-2
C′
x
-2
c) Reflect A′B′C′ over the x-axis.
Label the new coordinates A″, B″, and C″.
2.
2
-4
-6
-8
a) Draw the lines of symmetry for each figure.
Then, write the number of lines of symmetry.
Figure
Draw Lines of Symmetry
Number of Lines
of Symmetry
Equilateral triangle
Square
Regular pentagon
Regular hexagon
b) What pattern do you notice? _____________________________________________________
3.
How many lines of symmetry does each figure have?
a)
b)
1.2 Warm Up ● MHR 15
Name: _____________________________________________________
Date: ______________
1.2 Rotation Symmetry and Transformations
centre of rotation
● the point about which an object or a design is turned or rotated
rotation symmetry
when a shape or design can be turned about its centre of
rotation so it fits onto itself more than once in 1 complete turn
●
Link the Ideas
Working Example 1: Find Order and Angle of Rotation
centre of rotation
order of rotation
● the number of times a shape fits onto itself in 1 complete turn
angle of rotation
the smallest measure of the angle needed to turn a shape onto itself
● measured in degrees (°) or fractions of a turn
● angle of rotation = 360° ÷ order of rotation
●
For each shape, state the order of rotation and the angle of rotation.
Write the angle of rotation in degrees and as a fraction of a revolution.
a)
Solution
‰ Trace the shape onto a piece of tracing paper.
‰ Place your copy over the original.
‰ Rotate it to see the order and angle of rotation.
Order of rotation = 2, because when you rotate the
flag 1 complete turn it fits on itself 2 times.
Angle of rotation in degrees =
360°
2
=
°
1 turn
2
1
= turn
2
Angle of rotation as a fraction of a turn =
order of rotation
16
MHR ● Chapter 1: Symmetry and Surface Area
6
A revolution is
1 complete turn.
4
Name: _____________________________________________________
Date: ______________
b)
Solution
‰ Trace the shape onto a piece of tracing paper.
‰ Place your copy over the original.
‰ Rotate it to see the order and angle of rotation.
Order of rotation = 5.
When you rotate the pentagon 1 complete turn, it fits on itself
Angle of rotation in degrees =
=
times.
360°
5
°
Angle of rotation as a fraction of a turn =
=
1 turn
order of rotation
1
turn
1.2 Rotation Symmetry and Transformations ● MHR 17
Name: _____________________________________________________
Date: ______________
c)
Solution
‰ Trace the shape onto a piece of tracing paper.
‰ Place your copy over the original.
‰ Rotate it to see the order and angle of rotation.
Order of rotation = 1.
When you rotate the kite 1 complete turn, it fits on itself
time.
Angle of rotation in degrees = 360°
When you turn it about its centre of rotation it does not fit onto itself more
than once in 1 complete turn.
Angle of rotation as a fraction of a turn = 1 turn
This figure does not have
rotational symmetry.
Complete the chart.
Figure
a)
Order of
Rotation
Angle of Rotation
(Degrees)
360°
=
2
b)
c)
18
MHR ● Chapter 1: Symmetry and Surface Area
°
Angle of Rotation
(Fraction of a Turn)
1 turn
1
= turn
order of rotation 2
Name: _____________________________________________________
Date: ______________
Working Example 2: Relating Symmetry to Transformations
Look at the figures.
Figure 1
Figure 2
Figure 3
a) Complete the chart for each figure.
Solution
Type of
Symmetry
Number and
Direction of
Lines of
Symmetry
Figure 1
Figure 2
Figure 3
rotation
line
rotation and line
No lines of
symmetry
Number of lines
of symmetry:
of symmetry:
Direction:
Direction:
(↕)
●
Order of
Rotation
Angle of
Rotation
Number of lines
3
360°
=
3
●
1
●
1
1
°
2
°
360°
=
°
1.2 Rotation Symmetry and Transformations ● MHR 19
Name: _____________________________________________________
b) What shape is used to make each design?
How is each design made?
Date: ______________
Use translation,
reflection, rotation.
Solution
●
Figure 1 is made from 1 arrow. Rotate the arrow
the centre of
●
1
of a turn about
3
.
Figure 2 is made from 1
.
Figure 1
●
Translate the circle
times.
●
Figure 3 is made from 1
.
Reflect the hexagon both vertically (↕) and
Figure 2
Translate the hexagon
times.
(↔).
or
Figure 3
Figure 3
Look at each figure.
a) Circle the type of symmetry that each figure shows.
Figure A: LINE SYMMETRY or ROTATION
SYMMETRY or BOTH
Figure A
Figure B
Figure B: LINE SYMMETRY or ROTATION
SYMMETRY or BOTH
b) For Figure A, draw the line(s) of symmetry.
c) What is the order of rotation? Figure A:
Figure B:
d) How can you make each design using part of the figure and translations, reflections, or rotations?
Figure A: __________________________________________________________________
Figure B: __________________________________________________________________
20
MHR ● Chapter 1: Symmetry and Surface Area
Name: _____________________________________________________
Date: ______________
Check Your Understanding
_____________________________________________________
_____________________________________________________
A
A
Describe rotation symmetry. Use the design to help you.
A
1.
A
Communicate the Ideas
_____________________________________________________
2.
Maurice says the design shows rotation symmetry.
Claudette says the design shows line symmetry.
Who is correct? Circle MAURICE or CLAUDETTE or BOTH.
Give 1 reason for your answer.
_____________________________________________________
_____________________________________________________
Practise
3.
Each shape has rotation symmetry.
Figure 1
Figure 2
a) Mark the centre of rotation in each figure with a black dot.
b) Complete the chart.
Order of
Rotation
Figure 1
Angle of Rotation
(Degrees)
360°
=
4
°
Angle of Rotation
(Fraction of a Turn)
1 turn
=
order of rotation
1
turn
Figure 2
1.2 Rotation Symmetry and Transformations ● MHR 21
Name: _____________________________________________________
4.
Complete the chart.
Figure
Date: ______________
Use tracing paper to help you.
Rotation Symmetry
(Yes or No)
Order of Rotation
Angle of Rotation
(Degrees)
a)
b)
c)
5.
XOX
Each design has line and rotation symmetry.
Find the number of lines of symmetry and the order of rotation.
a)
b)
Number of lines of symmetry =
Number of lines of symmetry =
Order of rotation =
Apply
6.
Complete the drawing of the rotation.
Use the centre of rotation.
The order of rotation is 2.
E
22
MHR ● Chapter 1: Symmetry and Surface Area
Order of rotation =
Name: _____________________________________________________
7.
Date: ______________
Look at the design.
a) Draw the shape used to make the design.
b) Describe how to make this design using 1 or more of the following:
● translations
● reflections
● rotations
_____________________________________________________________________________
_____________________________________________________________________________
8.
a) Describe the symmetry shown on the playing card.
________________________________________________________
________________________________________________________
b) Why do you think the card is designed like this?
________________________________________________________
c) Does this card have line symmetry? Circle YES or NO.
Give 1 reason for your answer.
________________________________________________________
9. Part of each object has rotation symmetry of order 6.
Draw 2 other objects that have rotation symmetry of order 6.
1.2 Rotation Symmetry and Transformations ● MHR 23
Name: _____________________________________________________
Date: ______________
Math Link
Your company asks you to create a design using rotation symmetry.
a) Follow the steps to create this design with a rotation symmetry of 45°.
‰ Use tracing paper to copy the diamonds at the top and bottom of the design.
‰ Mark the centre.
‰ Draw a line of symmetry through the 2 diamonds and the centre.
This will be your design template.
‰ Copy your tracing onto another piece of paper.
Set a protractor on the centre of your diagram and mark points
at 45°, 90°, and 135°.
‰ Set your template behind the second piece of paper along the line
of symmetry.
‰ Rotate it to 45°. Copy the design.
‰ Rotate it to 90°. Copy the design.
‰ Rotate it to 135°. Copy the design.
On a separate sheet of paper, create a design of your own.
Your design must have rotational symmetry.
You will use this design for the cover of a box to hold your playing cards.
24
MHR ● Chapter 1: Symmetry and Surface Area
90°
135°
45°
Name: _____________________________________________________
1.3 Warm Up
1.
Date: ______________
What are the shapes of the faces?
Use a formula to find the surface area of each right prism.
a)
6m
10 m
4m
Area of front or back :
Area of top or bottom :
Area of end :
A=
A=
A=
SA = (2 × area of front or back) + (2 × area of top or bottom) + (2 × area of end)
= (2 ×
b)
) + (2 ×
=
+
=
m2
) + (2 ×
)
+
5 cm
10 cm
SA = 2 × (π × r2) + (π × d × h)
2.
Find the area of the shaded region.
10 cm
13 cm
Shaded area = area of large rectangle – area of small rectangle
5 cm
40 cm
=(
×
=
–
=
cm2
)–(
×
)
1.3 Warm Up ● MHR 25
Name: _____________________________________________________
Date: ______________
1.3 Surface Area
Link the Ideas
surface area
● the sum of the areas of all the faces of an object
The formula for the surface area of a rectangular prism and a cylinder is the same:
Perimeter = distance around a figure
SA = 2 × (area of base) + (perimeter of base) × (height)
1 cm
Surface area of rectangular prism:
SA = 2 × (area of base) + (perimeter of base) × (height)
= 2 × (4 × 2) + (4 + 2 + 4 + 2) × 1
= 2 × 8 + 12 × 1
= 16 + 12
= 28 cm2
4 cm
2 cm
Use the order of operations.
Surface area of cylinder:
SA = 2 × (area of base) + (perimeter of base) × (height)
= 2 × (π × r2) + (2 × π × r) × h
= 2πr2 + 2πrh
r
h
Working Example 1: Calculating Surface Area of a Solid
All the angles in the solid are right angles.
top
front
8 cm
side (right)
8 cm
24 cm
15 cm
8 cm
16 cm
16 cm
20 cm
front
24 cm
20 cm
Dimensions are the
length, width, and height.
a) What are the dimensions of the cutout piece?
Solution
The cutout piece is a right rectangular prism.
8 cm
8 cm
The dimensions of the piece are 8 cm by 8 cm by
26
MHR ● Chapter 1: Symmetry and Surface Area
16 cm
.
Name: _____________________________________________________
Date: ______________
b) What is the total surface area of the solid?
8(
rig
)
Solution
7
Find the area of the 9 faces. Number the faces to help keep track of them.
View
8 cm
1
15 cm
8 cm
20 cm
8 cm
2
24 cm
)
Shaded area = area of large rectangle – area of small rectangle
= (20 × 24) – (8 × 16)
=
–
=
cm2
cm2
=
Area = 8 × 8
8 cm
8 cm
cm2
=
8 cm
6
6
Area = 8 × 16
8 cm
15 cm
9(
Shaded area = area of large rectangle – area of square
= (15 × 20) – (8 × 8)
= 300 – 64
= 236 cm2
16 cm
16 cm
4
e)
Surface Area Calculation
20 cm
3 and 5
sid
4
5
1
bot
tom
ht
2
3
Method 1: Find the Surface Area of Each Face
Face
ck
(ba
Shaded area = area of large rectangle – area of small rectangle
= (15 × 24) – (8 × 16)
16 cm
=
–
=
cm2
24 cm
Area = 15 × 24
7
15 cm
=
cm2
24 cm
Area = 15 × 20
15 cm
=
8
cm2
20 cm
Area = 20 × 24
9
20 cm
=
cm2
24 cm
1.3 Surface Area ● MHR 27
Name: _____________________________________________________
Date: ______________
Total SA = 236 + 352 + 128 + 64 + 128 + 232 + 360 + 300 + 480
cm2
=
cm2.
The total surface area is
Method 2: Use Symmetry
8(
rig
k)
7
By symmetry, opposite faces match.
face 2 + face 5 = face 9 (bottom)
face 1 + face 4 = face 8 (right side)
c
(ba
5
1
bot
tom
sid
e)
4
3
9(
ht
2
6
)
face 6 + face 3 = face
(back)
So, calculate the area of faces 9, 8, and 7.
face 9 (bottom): 20 × 24 =
face 8 (right side): 15 × 20 =
face 7 (back): 15 × 24 =
Total of 3 faces =
=
+
+
cm2
To find the total surface area, double the area for face 9 + face 8 + face 7.
1140 × 2 =
The total surface area is
cm2.
Did You Know?
If you cut a right rectangular piece out of 1 corner of
a rectangular prism, the surface area does not change.
Figure 1
The surface area does change if the cutout goes across
the solid.
Figure 3
28
MHR ● Chapter 1: Symmetry and Surface Area
Figure 2
Name: _____________________________________________________
Date: ______________
e
Fac
6
A set of concrete steps has these dimensions.
Fac
e
a) Calculate the area of the faces that are not against the ground.
Draw each face and label the measurements:
Face 1:
Fac
e
Fac
e
Face 3:
Face 5:
Face 6:
7
4
60 cm
40 cm
2
Fac
e
Face 2:
3
Fac
e
1 2
0c
m
80 cm
e5
Fac
90 cm
80 cm
20 cm
Face 4:
Face 7:
60 cm
40 cm
20 cm
90 cm
Find the area of faces 1 to 7:
Find the total area:
Sentence: __________________________________________________________________
b) What is the area of the face of the step against the ground?
Sentence: __________________________________________________________________
1.3 Surface Area ● MHR 29
Name: _____________________________________________________
Date: ______________
Working Example 2: Painting a Bookcase
22 cm
Raubyn wants to paint the shelves and the outside of the bookcase.
He will not paint the back.
The back is very thin plywood. It does not add any extra depth.
115 cm
a) What assumptions could you make about how he calculates
the area to be painted?
140 cm
Solution
Assumptions could include:
●
●
●
●
He paints the underside of
shelves.
The measurement of the shelves is the same as the top.
He does not paint the area of the base on which the bookcase stands.
He paints the bookcase after it is put together.
Literacy Link
An assumption is something
you have not been told, but
you believe to be true.
b) Find the surface area that Raubyn needs to paint.
Solution
Group similar size surfaces together. Draw and label the dimensions of each group.
Group 1: Both sides of top + top and bottom of each shelf = 8 rectangular areas
Surface area = 8 × area of top or bottom of shelf
=8×l×w
= 8 × 115 × 22
115 cm
22 cm
cm2
=
Group 2: Both sides of the 2 ends of the bookcase
22 cm
Surface area = 4 × area of side
=4×l×w
= 4 × 22 ×
Side 140 cm
cm2
=
Total surface area = 20 240 + 12 320
=
cm2
The surface area Raubyn needs to paint is
30
cm2.
MHR ● Chapter 1: Symmetry and Surface Area
Name: _____________________________________________________
Calculate the outside surface area of the building.
Do not include the bottom of the building.
Date: ______________
5m
4m
Label the dimensions of similar surfaces.
5m
10 m
6m
2 triangles:
4m
2 rectangles (front and back):
4m
6m
6m
2 rectangles (sides):
2 rectangles (roof ):
Find the area of each group.
Total surface area:
Sentence: ____________________________________________________________________
1.3 Surface Area ● MHR 31
Name: _____________________________________________________
Date: ______________
Check Your Understanding
Communicate the Ideas
1.
Use 24 interlocking blocks to build 2 different solid objects.
a) Explain how symmetry helps you find the surface area of 1 of your objects.
Group matching
surfaces.
____________________________________________________________________________
____________________________________________________________________________
b) Slide the 2 objects together. What is the area of overlap between the objects?
area where 2 objects touch
Sentence: ____________________________________________________________________
c) How does the overlap affect the total surface area of your composite object?
____________________________________________________________________________
____________________________________________________________________________
2.
Nick makes a 2-layer cake. He puts strawberry jam between
the 2 layers.
He wants to cover the outside of the cake with chocolate icing.
Describe how he can find the area that needs icing.
_________________________________________________
_________________________________________________
_________________________________________________
_________________________________________________
32
MHR ● Chapter 1: Symmetry and Surface Area
Literacy Link
A composite object is an object
made from 2 or more separate
objects.
Name: _____________________________________________________
Date: ______________
Practise
3.
Each object is made with centimetre cubes.
Calculate the surface area. Include the bottom area.
a)
b)
Number the faces.
Area of each face:
Count the cubes
on each face.
Face 1:
Face 2:
Face 3:
Face 4:
Face 5:
Face 6:
Face 7:
Face 8:
Total surface area:
4.
a) If you build these cubes and slide them together, what is the area of the overlap?
Assume the dots are 1 cm apart.
b) What is the surface area of the composite solid?
Number the sides.
1.3 Surface Area ● MHR 33
Name: _____________________________________________________
5.
Date: ______________
In the solid, all angles are right angles.
right side
9 cm
front
top
25 cm
25 cm
14 cm
35 cm
18 cm
What are the dimensions of the cutout piece? Write them on the diagram.
length: 35 – 18 =
Look at the diagram
of the top.
width:
height: 25 –
Apply
6.
Use centimetre cubes to build this object.
=
Look at the diagram
of the front.
a) What is the surface area of the object?
1 cm
Sentence: ____________________________________________________________________
b) Using the same 10 cubes, build a rectangular prism.
Find the surface area of your rectangular prism.
Sentence: ____________________________________________________________________
c) Are the 2 surface areas the same? Circle YES or NO.
Give 1 reason for your answer.
____________________________________________________________________________
7.
List 3 places or situations where surface area is important.
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
34
MHR ● Chapter 1: Symmetry and Surface Area
Name: _____________________________________________________
8.
Date: ______________
a) How can you use symmetry to find the surface area of this object?
_______________________________________________________
4 cm
8 cm
3 cm
_______________________________________________________
12 cm
b) Find the surface area of the outside of the object.
Area of front or back:
A = (l × w) – (π × r2)
Area of rectangle – area of circle
Area of top or bottom:
Area of inside of hole:
A = 2 × π × r × length of hole
Area of end of rectangular prism:
Total surface area:
Sentence: ____________________________________________________________________
Math Link
You are making a box to store the deck of playing cards you designed in Math Link 1.1.
a) How many cards are in your deck?
b) What are the dimensions of your deck of cards?
length:
c)
width:
height:
What is the surface area of your deck of cards?
Sentence: _______________________________________________________________________
1.3 Math Link ● MHR 35
Name: _____________________________________________________
Date: ______________
Graphic Organizer
Fill in the boxes.
Rotation
Symmetry
Line
Symmetry
Symmetry and
Surface Area
Definition:
Definition:
Surface Area
Example:
Example:
centre of rotation
Definition:
Define order of rotation:
Ways to find the surface
area of an object
•
Example: For the fan above, the
Give 1 example of each of the
lines of symmetry:
• vertical
• horizontal
order of rotation is
Define angle of rotation:
•
• oblique
You can complete a symmetrical
drawing by
Example: For the fan above, the
angle of rotation is
•
•
or
Example: For the fan above, the
fraction of a turn is
36
MHR ● Chapter 1: Symmetry and Surface Area
•
Name: _____________________________________________________
Date: ______________
Chapter 1 Review
For #1 to #6, choose the number that best matches the description.
1.
line symmetry
another name for a reflection line
2.
rotation symmetry
type of symmetry where the shape is divided
into reflected halves
3.
angle of rotation
the total area of all the faces of an object
4.
surface area
type of symmetry where a shape is turned onto itself
5.
line of symmetry
number of times a shape fits onto itself in 1 turn
6.
order of rotation
the size of turn for a shape to rotate onto itself
1.1 Line Symmetry, pages 6–14
7.
Draw the lines of symmetry. Write the number of lines of symmetry for each design.
Then, describe each line of symmetry using the terms vertical, horizontal, and oblique.
a)
b)
← Number of lines of symmetry →
← Description →
8.
Half of a figure is drawn. The dashed line is the line of symmetry. Finish drawing each figure.
a)
b)
Chapter 1 Review ● MHR 37
Name: _____________________________________________________
9.
a) Draw a reflection of the shape in the y-axis.
Label the image A′, B′, C′, D′, E′, and F′.
A
b) Write the coordinates of the reflection image.
B
A′ (
,
)
Date: ______________
y
D4
E
F
C
2
B′
C′
D′
E′
F′
-4
-2
2
4
x
-2
c) Using a different colour, translate the original shape R6, D3.
Label the image A″, B″, C″, D″, E″, and F″.
d) Which transformation shows symmetry? Circle REFLECTION or TRANSLATION.
Describe the symmetry using the terms vertical, horizontal, and oblique.
_____________________________________________________________________________
1.2 Rotation Symmetry and Transformations, pages 16–24
10. Complete the chart for each diagram.
Diagram
Order of
Rotation
Angle of Rotation
(Degrees)
Angle of Rotation
(Fraction of a Turn)
a)
360°
=
b)
11. What type of symmetry does the design have?
Circle ROTATION SYMMETRY or LINE SYMMETRY or BOTH.
Give 1 reason for your answer.
____________________________________________________________
____________________________________________________________
38
MHR ● Chapter 1: Symmetry and Surface Area
1
Name: _____________________________________________________
Date: ______________
1.3 Surface Area, pages 26–35
12. The triangular prism has 1 triangular end fastened to the wall.
All the other faces are showing.
What is the surface area of the faces that are showing?
7.8 cm
18 cm
9 cm
Sentence: ________________________________________________________________________
8 cm
13. Two blocks are placed 1 on top of the other.
a) If the blocks are separated, what is the surface area of
each block?
18 cm
3 cm
24 cm
4 cm
Small Block:
Area of front or back:
16 cm
Area of top or bottom:
Area of side:
Area of top or bottom:
Area of side:
Total surface area:
Large Block:
Area of front or back:
Total surface area:
b) What is the total surface area of the 2 blocks when separated?
Sentence: _____________________________________________________________________
c) What is the surface area of the stacked blocks?
(answer to part b) – (2 × area of
shape where blocks touch)
Sentence: _____________________________________________________________________
Chapter 1 Review ● MHR 39
Name: _____________________________________________________
Date: ______________
Key Word Builder
Use the clues to find the key words in Chapter 1. Then, write them in the crossword puzzle.
Across
6. the sum of the areas of all the faces of an object (2 words)
8. the measure of an angle to turn a shape onto itself (3 words)
Down
1. a type of symmetry where an object is divided into 2 identical halves by a line (2 words)
2. occurs when a shape can be turned about its centre of rotation and fit onto itself more than once
in a complete turn (2 words)
3. a line running through the centre of an object so that the halves on each side are mirror images
(3 words)
4. the number of times a shape or design fits onto itself in 1 complete turn (3 words)
5. the point about which the rotation of an object turns (3 words)
7. when a shape or design can fit onto itself by reflection or rotation, it has
(1 word)
40
MHR ● Chapter 1: Symmetry and Surface Area
Name: _____________________________________________________
Date: ______________
Chapter 1 Practice Test
For #1 to #4, choose the best answer.
1.
2.
3.
Which design has rotation symmetry of order 2?
A
B
C
D
6119
How many lines of symmetry does the design have?
A 0
B 1
C 2
D 4
Which figure does NOT have rotation symmetry?
A
B
C
D
OHO
Complete the statements in #4.
4.
The design has rotation symmetry.
a) The order of rotation is
b) The angle of rotation is
Chapter 1 Practice Test ● MHR 41
Name: _____________________________________________________
Date: ______________
Short Answer
5.
Use these upper-case letters:
a) List 5 letters that have horizontal line symmetry.
b) List 5 letters that have vertical line symmetry.
c) List 1 letter that has oblique line symmetry.
d) List 5 letters that have rotation symmetry where the angle of rotation is 180°.
6.
a) What is the surface area of the rectangular prism?
12 cm
7 cm
5 cm
b) The rectangular prism has a 1-cm cube cut out of 4 of its corners.
How many faces of each cube are visible before they are removed?
How many faces are visible after they are removed?
What is the surface area of this rectangular prism with 4 of its corners removed?
42
MHR ● Chapter 1: Symmetry and Surface Area
12 cm
7 cm
5 cm
Name: _____________________________________________________
Date: ______________
Math Link: Wrap It Up!
You employer wants to see the playing card box that you designed in the Math Links from this chapter.
‰ Include the design from Math Link 1.1, page 14.
‰ Describe the type of line symmetry your card design shows.
Use the terms horizontal, vertical, and oblique.
________________________________________________________________________________
‰ Include the design from Math Link 1.2, page 24. This will be the design for the cover of your box.
‰ Write the dimensions and the surface area of the box needed to hold 1 set of your playing cards.
Use your answers from Math Link 1.3, page 35.
________________________________________________________________________________
‰ Write the total surface area of 6 boxes of cards.
SA of 1 box:
Total SA of 6 boxes:
‰ Your company wants to sell a large package containing 6 decks of your playing cards.
Draw how you would arrange the decks for packaging.
‰ Using your dimensions for 1 box of cards, calculate the measurements for your large package.
Write them on your design.
‰ Calculate the surface area of your large package.
Math Link: Wrap It Up! ● MHR 43
Name: _____________________________________________________
Date: ______________
Challenge
Making a Paper Airplane
Materials
• paper
• ruler
• scissors
1.
Make a paper airplane by following the folding instructions.
In Step 6, make 4 small cuts as shown to create tabs. Fold the tabs up to make flaps.
Step 1
2.
Step 2
Step 3
Step 4
Step 5
a) Using your airplane from #1, find the total area of the top view of the wings. Include the flaps.
Measurement of wings of airplane:
Area of triangle (2 wings):
b) Fly the airplane 5 times. Record the distance travelled in metres.
Record the direction travelled in each flight.
Flight Distance Travelled (m)
1
2
3
4
5
44
Step 6
MHR ● Chapter 1: Symmetry and Surface Area
Direction of Flight
Name: _____________________________________________________
3.
Design and create a non-symmetrical airplane.
Date: ______________
Literacy Link
Non-symmetrical means it is
not the same on both sides.
a) Find the new surface area of the wings.
Drawing of wings (include measurements):
Area of 2 wings:
Sentence: _____________________________________________________________________
b) Fly the airplane 5 times. Record the distance travelled, in metres.
Record the direction travelled in each flight.
Flight Distance Travelled (m)
Direction of Flight
1
2
3
4
5
4.
Which airplane flew the best? Circle SYMMETRICAL or NON-SYMMETRICAL.
Give 2 reasons why. Include the words surface area and symmetry in your explanation:
Reason 1 (surface area): ____________________________________________________________
Reason 2 (symmetry): ______________________________________________________________
Challenge ● MHR 45
Answers
Get Ready, pages 2–3
1.1 Line Symmetry, pages 6–14
1. a) 4, vertically up b) 3 units horizontally left, 4 units vertically down
2. a)
y
b)
y
rr
Working Example 1: Show You Know
a) 1; horizontal b) 1; vertical c) 4; 1 horizontal, 1 vertical, 2 oblique
d) 1; vertical
Working Example 2: Show You Know
line of reflection
x
r
a)
b)
x
3. a)
D
G
E
b) D′ = (2, 3) E′ = (–1, 3)
F′ = (–1, 2) G′ = (2, 2)
y
E’
D’
Communicate the Ideas
F’
G’
2. KENDRA. It does not overlap itself if it’s cut out and folded.
1. NO. A rectangle that is a square has 4 lines of symmetry.
Practise
x
3. a)
b)
c)
d)
4. a)
b)
4. 286 cm2
Math Link
1. a) YES b) 3
2. a) YES b) NO
3. a)
b) C
C
c) C′
D
D'
E
E'
r
d) vertically
C'
4. a) A to A′: 5, 3; B to B′: 5 spaces right and 3 spaces up; C to C′: 5 spaces
right and 3 spaces up
b) Answers will vary. Example: 4 spaces left and 4 spaces down would
put the translation in quadrant III.
1.1 Warm Up, page 5
1. a)
y
B
4
C
A
D
E
2
F
0
A‘
b)
2
E‘
B‘
O‘
G‘
2
6. a)
b)
c)
d)
A′ (0, –1); B′ (1, –2); C′ (3, –2); D′ (3, –1);
E′ (2, –1); F′ (1, 0)
F‘
–2
5.
4 x
D‘
C‘
y
O
2
D′ (0, 0); O′ (–2, 2); G′ (–2, 1)
G
D‘ D
2 x
0
Apply
–2
7. a)
b) A′ (–3, 6); C′ (–2, 2); E′ (–5, 3)
y
c) YES; The line of reflection
divides the original image
and the reflection image into
2 identical parts.
A’
2. a, b, d
3. a)
5
b)
4 cm
E’
3 cm
C’
4. Answers will vary.
46
MHR ● Chapter 1: Symmetry and Surface Area
-5
0
x
8. a) 1 b) 1 c) 2 d) 4
Working Example 2: Show You Know
9. a) B, C, D, E, H, I, K, O, X b) A, H, I, M, O, T, U, V, W, X, Y
c) H, I, O, X
10. Answers will vary: Examples:WOW W WHO W HAH H HI HI
O
A
H
O
W
H
a) Figure A: BOTH; Figure B: ROTATION SYMMETRY
b)
11. Answers will vary. Example:
88
c) Figure A: 2; Figure B: 4 d) Figure A: reflection along a horizontal or
vertical line; Figure B: rotation about the centre
Communicate the Ideas
Math Link
Answers will vary.
1. Answers will vary. Example: Rotation symmetry is when a section of
a design can be rotated around the centre and shows symmetry with
other parts.
1.2 Warm Up, page 15
y B’
1. a) and c) B
b) A′ (1, 1); B′ (1, 9); C′ (4, 1)
2. BOTH; The design can be folded over a horizontal or vertical line of
symmetry to overlap itself, or a corner piece can be rotated 4 times to
create the design.
Practise
3. a)
A
C
A’
C’
A’’
x C’’
b)
Order of
Rotation
B’’
2. a)
Figure
Lines of Symmetry
Figure 1
4
360°
= 90
4
°
Figure 2
2
180°
Number of Lines
of Symmetry
Equilateral
triangle
Angle of
Rotation
(Degrees)
Angle of Rotation
(Fraction of a Turn)
1 turn
1
= turn
order of rotation 4
1
turn
2
3
4.
Rotation
Symmetry
(Yes or No)
Order
of
Rotation
Angle of
Rotation
(Degrees)
a)
Yes
4
90°
b)
Yes
3
120°
Yes
2
180°
Square
Figure
4
Regular pentagon
5
Regular hexagon
6
b) The number of sides is equal to the number of lines of symmetry.
3. a) 5 b) 0
1.2 Rotation Symmetry and Transformations, pages 16–24
Working Example 1: Show You Know
Order of
Rotation
Angle of Rotation
(Degrees)
Angle of Rotation
(Fraction of a Turn)
a)
2
360°
= 180°
2
1 turn
1
= turn
order of rotation 2
b)
1
360°
1 turn
c)
3
120°
1 turn
1
= turn
order of rotation 3
c)
XOX
5. a) Number of lines of symmetry = 6, Order of rotation = 6 b) Number of
lines of symmetry = 2, Order of rotation = 2
Answers ● MHR 47
Apply
Apply
6.
6. a) The surface area is 36 cm2. b) The surface area of a 1 × 2 × 5
rectangular prism is 34 cm2. c) NO. There are more surfaces that
overlap in the 1 × 2 × 5 rectangular prism.
E
7. Answers will vary. Examples: when icing a cake, painting a building, or
wrapping a present.
E
7. a)
8. a) The top and bottom, left and right, and front and back are symmetrical
so they have the same surface area.
; square
b) The surface area is 324.6 cm2.
b) Rotate the figure 180° and join the 2 figures. Translate the new figure
to the right so it does not overlap. Join the two figures. Now, join the
new figure with the original one on the right.
Math Link
Answers will vary. Examples: a) 52 b) length: 10 cm; width: 7 cm; height:
2 cm c) The surface area is 208 cm2.
Graphic Organizer, page 36
Rotation Symmetry
8. a) The top half is rotated 180°. b) Cards are designed so they can be
read while being held from either end. c) NO; If you fold it in half,
the design does not overlap.
9.
Definition: when a shape or design can be turned about its centre of rotation
so it fits onto itself more than once in 1 complete turn
Define order of rotation: the number of times a shape fits onto itself in 1
complete turn
Example: 8
Define angle of rotation: the minimum measure of the angle needed to turn
a shape or design onto itself
Example: 45º
1
Example: turn
8
Surface Area
Math Link
Definition: the number of square units needed to cover an object
Answers will vary.
1. a) 248 m2 b) 471 cm2
Ways to find the surface area of an object: find the surface area of each face
and add them together; use symmetry; for a composite shape, find the
surface area of each part and subtract the area of the overlapping parts
2. 470 cm2
Line Symmetry
1.3 Warm Up, page 25
1.3 Surface Area, pages 26–35
Working Example 1: Show You Know
a) The total area of the faces that are not against the ground is 19 600 cm2.
b) The area of the step that is against the ground is 7 200 cm2.
Definition: a type of symmetry where a line of symmetry divides an image
or object into 2 identical, reflected halves
Example: the Canadian flag has line symmetry
Examples:
• vertical line of symmetry:
• horizontal line of symmetry:
Working Example 2: Show You Know
The surface area is 284 m2.
Communicate the Ideas
1. Answers will vary. Example:
a)
• oblique line of symmetry:
You can use symmetry to find the surface area of 1 side, and then you
can multiply it by 2 to get the surface area of the opposite side.
b)
2
The area of overlap is 12 cm .
You can complete a symmetrical drawing by: paper folding; measuring;
counting
Chapter 1 Review, pages 37–39
c) The overlap reduces the total surface area by twice the amount of the
surface area that is touching the other shape.
1. type of symmetry in which the shape can be divided into reflected halves
2. type of symmetry in which a shape is turned onto itself
2. Nick should find the area of each shape and add them together. Then he
should subtract 2 times the area of the overlap (the bottom of the
cylinder). He should also subtract the area of the bottom.
3. the size of turn for a shape to rotate onto itself
Practise
6. number of times a shape fits onto itself in 1 complete turn
3. a) 72 cm2 b) 112 cm2
4. a) The area of overlap is 4 cm2. b) The surface area is 70 cm2.
5. length = 17 cm; width = 9 cm; height = 11 cm
48
MHR ● Chapter 1: Symmetry and Surface Area
4. the total area of all the faces of an object
5. another name for a reflection line
7. a)
10.
Number of lines of symmetry: 4
Description: 1 horizontal, 1 vertical, 2 oblique
b)
Number of lines of symmetry: 6
Description:1 horizontal, 1 vertical, 4 oblique
Order
of
Rotation
Angle of
Rotation
(Degrees)
Angle of Rotation
(Fraction of a
Turn)
a)
4
360°
= 90°
4
1
4
b)
8
45°
1
8
Diagram
11. ROTATION SYMMETRY; There is no line symmetry because of the
shading and overlapping.
12. The total surface of the faces that are showing is 521.1 cm2.
8. a)
13. a) small block: 444 cm2; large block: 1088 cm2 b) The surface area of
the 2 blocks when separated is 1532 cm2. c) The surface area of the
stacked blocks is 1244 cm2.
b)
Key Word Builder, page 40
Across:
6. surface area 8. angle of rotation
Down:
9. a) and c)
y
D4
E
A
D‘
F
F‘
B
C
2
A‘
Chapter 1 Practice Test, pages 41–42
1. D 2. D 3. D
C‘
-2
F‘‘
2
B‘‘
B‘
D‘‘
E‘‘
A‘‘
-4
1. line symmetry 2. rotation symmetry 3. line of symmetry
4. order of rotation 5. centre of rotation 7. symmetry
E‘
4
x
C‘‘
-2
4. a) 8 b) 45°
5. Answers will vary. Examples: a) B, C, D, E, H, I, K, O, X
b) A, H, I, M, O, T, U, V, W, X, Y c) O d) H, I, N, O, S, X, Z
6. a) 358 cm2 b) 3; 3; 358 cm2
Math Link: Wrap It Up!, page 43
Answers will vary.
b) A′ (5, 3) B′ (5, 2) C′ (1, 2) D′ (1, 4) E′ (3, 4) F′ (3, 3)
d) REFLECTION; The reflection shows a vertical line of symmetry.
Challenge, page 44–45
Answers will vary.
Answers ● MHR 49