Math 6
Unit 3
Less n 11
Perimeter of Polygons
Regular Polygons in Space
The Mars Exploration Rover A (MER A) was launched on
June 5, 2003, followed by the launch of the MER B on
June 25, 2003. They both landed on Mars in January of
2004. They were designed to roam the surface of Mars
for 90 days but they were still operating as of August
2010. Their job is to gather data and photos to send
back to Earth. This allows scientists to study the
geology of the planet.
A major question that scientists would like to answer
is, “Was there ever water on the surface of Mars?”
They can find this by observing the rocks, minerals
and geological formations that are on the surface.
The MER B discovered some interesting
symmetrical structures on the way to the
Victoria crater. The stones were shaped
like regular polygons.
Reflection
What are some characteristics of a regular polygon?
Math 6
11-1
Math 6
Unit 3
Lesson 11: Perimeter of Polygons
Objectives for this Lesson
In this lesson you will explore the following concepts:
• Explain how the perimeter of any polygon can be determined
• Generalize a rule for finding the perimeter of rectangles and squares
• Write and explain the formula for finding the perimeter of any given
rectangle (using patterns and variables)
• Solve a given problem involving the perimeter of polygons
Perimeter of Regular Polygons
The perimeter of a polygon is found by adding the measures of each side.
You should recall finding the perimeter of figures with side lengths marked
on the figure.
Example 1
Find the perimeter of the playground.
To find the perimeter, add
the length of all of the sides together.
9.2 m
4.5 m
8.1 m
8.1 + 9.2 + 4.5 + 8.6 + 12 = 42.4
The perimeter of the playground is
42.4 metres.
12 m
8.6 m
You should be able to make a rule for finding the perimeter of a regular polygon.
11-2
Math 6
Unit 3
Lesson 11: Perimeter of Polygons
Example 2
Find the perimeter of a regular hexagon with side length of 12 metres.
Record the number of sides, n, and the side length, s: n = 6, s = 12
Use the rule:
P=n•s
P = (6) • (12)
P = 72 metres
The perimeter formula for a regular polygon should help you to find missing
side lengths as well as the perimeter.
Example 3
A regular pentagon has a perimeter of 30 centimetres.
What is the length of each side?
Write the parts of the rule that you know:
P = 30, n = 5, s = ?
Use the rule:
P=n•s
(30) = (5) • s
Divide both sides by 5:
30 5 : s
=
5
5
Write the units on your answer:
6 cm = s
Perimeter of Rectangles
A special type of quadrilateral is a rectangle.
The Properties of a Rectangle are:
• Opposite sides are congruent
w
• All angles are right angles
l
You should be able to find a rule for the perimeter of a rectangle when given
the length and width.
11-3
Math 6
Unit 3
Lesson 11: Perimeter of Polygons
Let’s Explore
Example 4
Find the perimeter of a rectangle with length of 4.7 cm and width of 2.5 cm.
Write the parts of the rectangle you know:
l = 4.7, w = 2.5
Use the rule:
P = 2l + 2w
Use parentheses to show multiplication:
P = 2(4.7) + 2(2.5)
Simplify:
P = 9.4 + 5
Write the unit on your answer:
P = 14.4 cm
Math 6
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Math 6
Unit 3
Lesson 11: Perimeter of Polygons
Solving Perimeter Problems
You should be ready to use your formulas for finding perimeter of regular
polygons and rectangles. Solve the problems using your formulas.
Example 5
Alyssa is putting a border in her room. The walls in her room form a rectangle.
The length is 10 metres and the width is 12 metres. The border comes in
packages of 6 metres. How many packages of border will she need to buy?
Find the perimeter of the room:
P = 2(10) + 2(12) = 20 + 24 = 44 m
Make a model:
Perimeter
of room
6 metres
in a roll
Number
of rolls
Solve: 44 ÷ 6 ≈ 7.3
Analyze the answer:
The answer indicates that Alyssa will need more than 7 rolls.
Since the border is sold by the roll, she will need 8 rolls of border.
10 m
11-5
12 m
Math 6
Unit 3
Lesson 11: Perimeter of Polygons
Example 6
Daksha and Lian are playing a game that has a board shaped like a regular
hexagon. The perimeter of the board is 54 centimetres. What is the length of
each side of the game board?
Write the parts of the perimeter rule
that you know:
Use the rule:
P = 54, n = 6, s = ?
P=n•s
(54) = (6) • s
Divide both sides by 6:
54 6 : s
=
6
6
Write the unit on the answer:
9 cm = s
Each side of the hexagon game board is 9 cm long.
Math 6
11-6
Math 6
Unit 3
Lesson 11
Perimeter of Polygons
Let’s Explore
Exploration 1: Perimeter of Regular Polygons
Materials: Pencil
Find the perimeter of the regular polygons with a side length of 3 cm.
Regular Polygon
Number of Sides
Side Length (cm)
Perimeter (cm)
Equilateral Triangle
3
3
9
Square
4
3
12
Pentagon
5
3
Hexagon
Heptagon
Octagon
Nonagon
Decagon
1.
What pattern do you notice in the perimeters of the regular polygons?
2.
How would the pattern change if the side length is 4 cm?
Math 6
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Math 6
Unit 3
Lesson 11: Perimeter of Polygons
3.A regular polygon has 15 sides. The side length is 4 units.
What is the perimeter?
4.
Write a rule to find the perimeter, P, of a regular polygon with n sides
and a side length of s.
Let’s Explore
Exploration 2: Perimeter of a Rectangle
Materials: String, Centimetre Ruler, Scissors, Pencil
1. Cut a length of string that is 42 centimetres long.
2. C
reate at least six rectangles that have whole number side lengths. Record the
measures in the given table.
Length
11-8
Width
Perimeter
Math 6
Unit 3
Lesson 11: Perimeter of Polygons
3. Do you notice a relationship between the side lengths and the perimeter?
4. W
rite a rule for the perimeter, P, of a rectangle given the length, l,
and the width, w.
Let’s Practice
For 1 – 18: Find the perimeter or the missing measure of each polygon.
1.
3m
2.
s = 12 cm
2.4 m
2.4 m
5m
3.
4.
10.1 mm
156 cm
16.4 mm
187 cm
9.8 m
5.
6.
a
6.3 m
P = 52.3 m
14.8 m
Math 6
12.9 m
P = 11 m
w
3.4 m
11-9
Math 6
Unit 3
7.
Lesson 11: Perimeter of Polygons
3 mm
8.
32 cm
3 mm
1.5 mm
9 mm
54 cm
8 mm
9 mm
9.
15 m
10.
25 m
P = 108 cm
20 m
11. regular octagon, side length = 18 m
12. square, side length = 3.75 mm
13. nonagon, perimeter = 58.5 m
11-10
Math 6
Unit 3
Lesson 11: Perimeter of Polygons
14. rectangle, l = 14 cm, w = 12 cm
15. rectangle, l = 10 m, P = 32 m
16. equilateral triangle, side length = 8 mm
17. rectangle, w = 18 cm, P = 76 cm
18. regular pentagon, side length = 3.5 cm
Math 6
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Math 6
Unit 3
Lesson 11: Perimeter of Polygons
For 19 – 22: Use order of operations to find the perimeters of the given
rectangles in three ways.
Example: l = 3, w = 2
19.
l = 3.5, w = 2.4
20.
l = 10, w = 8
21.
l = 7, w = 2.5
22.
l = 52, w = 38
P=l+l+w+w
P = 2l + 2w
P=3+3+2+2
P = 10
P = 2(3) + 2(2)
P=6+4
P = 10
P = 2(l + w)
P = 2(3 + 2)
P = 2(5)
P = 10
23. R
eflect: What can you say about the three rules that you used in
problems 19 – 22?
For 24 – 27: Solve the problems using the rules for perimeter.
24. Z
ach and Nina each make a sign to put on the door of their bedroom.
Both signs have a perimeter of 60 centimetres. Zach’s sign is a rhombus.
Nina’s sign is a regular pentagon. How are the side lengths of their
signs different?
11=12
Math 6
Unit 3
Lesson 11: Perimeter of Polygons
25. C
ameron’s mom has fenced a garden beside their house.
The fence and the wall of the house form a rectangle.
What is the total length of the fence?
Wall of House
4m
10 m
26. A
lyssa and her dad want to build a dog run. They would like the dog run to
have a length of 18 metres and a width of 6 metres. What would be the
perimeter of the dog run if the length is only half of this measure?
27. E
ach square in the pattern has a side length that is half the one before it.
What is the perimeter of the whole figure?
12
Math 6
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Math 6
Unit 3
Lesson 11: Perimeter of Polygons
28. U
se a tape measure to find the perimeter of something large in your area.
Some possibilities include: a basketball court, tennis court, a playground
or a house. Draw a picture of the figure you measured and use a rule to
find the perimeter.
Mixed
Re vi ew
Number of Volunteers
For 1 – 5: Use the bar graph below that shows the number of parent
volunteers at Crestview Middle School.
100
90
80
70
60
50
40
30
20
10
0
1.
11-14
Parent Volunteers
80
65
45
45
Athletics
Band
86
72
55
30
2005
2006
2007
2008
How many more volunteers were there for athletics than band in 2008?
Math 6
Unit 3
Lesson 11: Perimeter of Polygons
2.
What was the total number of volunteers for 2006?
3.
How many more total volunteers were there in 2008 than in 2005?
4.
If the trend continues, will the number of volunteers increase or
decrease in 2009?
5.
What fraction of the volunteers in 2007 worked for band?
Math 6
11-15
Math 6
Unit 3
11-16
Lesson 11: Perimeter of Polygons