Chapter 1: Measurement - Mr. McDonald's Homepagemrmcdonaldshomepage.weebly.com/uploads/1/3/9/2/13921997/math_1201... · Chapter 1: Measurement Section 1.4 Page 12 Determining the

Chapter 1: Measurement Section 1.1

Page 1

Chapter 1: Measurement

Section 1.1: Imperial Measures of Length

Terminology:

SI System of Measurement:

The system is an abbreviation for Le Systeme International d’Unites. Since 1960,

this form of the metric system has been adopted by many countries, including

Canada.

Imperial Units:

Measurements including the inch, foot, yard, and mile.

Referent:

A common item that can be used to approximate a given measurement.

The referents that we will be using include:

Metric System Imperial System

1mm the thickness of a dime/ thickness of a fingernail

1in

the distance from the tip of your thumb to your first knuckle/ width of a hockey

puck/ width of a loonie

1cm the width of a fingernail 1ft

the length of a floor tile/ the length of a person’s foot/ the distance from

your wrist to your elbow

1m the height from the floor to the doorknob of a standard

door 1yd

a person’s arm span/ the length of a guitar

1 km the distance one can walk comfortably in 15 minutes

1mi the distance a person can walk comfortably in 20

minutes


Page 2

Conversions within the Imperial System of Measure

Conversions:

Imperial Unit Abbreviation Conversions

Inch in.

Foot ft. 1 ft = 12 in

Yard yd. 1 yd = 3 ft

1 yd = 36 in

Mile mi. 1 mi = 1760 yd 1 mi = 5280 ft

Note: Within the imperial system, the smallest unit indicated on a ruler will be 1

16 in.

Converting Between Imperial Units

Example 1: Convert 5 yd to:

i) Feet

ii) Inches

Example 2: Convert 51 in to:

i) Feet and Inches

ii) Yards, Feet, and Inches


Page 3

Example 3: Convert 7 yd to:

iii) Feet

iv) Inches

Example 4: Convert 62 in to:

iii) Feet and Inches

iv) Yards, Feet, and Inches


Page 4

Solving a Problem Involving Converting Between Units

Example 1:

Anne is framing a picture. The perimeter of the framed picture will be 136 in.

(a) What will be the perimeter of the framed picture in feet and inches?

(b) The framing material is sold by the foot. It costs $1.89/ft. What will be the cost of

material before taxes?

Example 2:

Ben buys baseboard from a bedroom. The perimeter of the bedroom, excluding closets

and doorway, is 37 ft.

(a) What length of baseboard is needed, in yards and feet?

(b) The baseboard material is sold by the yard. It costs $5.99/yd. What is the cost of

material before taxes?


Page 5

Solving a Problem Involving Two Unit Conversions

Example 1:

The school council has 6 yd of fabric that will be cut into strips 5 in wide to make

decorative banners for the school dance.

(a) How many banners can be made?

(b) Use unit analysis to verify?

Example 2:

Tyrell has 4 yd of cord to make friendship bracelets. Each bracelet needs 8 in of cord.

(a) How many bracelets can Tyrell make?

(b) Use unit analysis to check the conversions.


Page 6

Solving a Problem Involving Scale Diagrams

Example 1:

A map of Alaska has a scale of 1: 4 750 000. The distance on the map between Paxson

and the Canadian border is 311

16 in. What is this distance to the nearest mile?

Example 2:

On the map with a scale of 1: 4 750 000, the distance between Seward and Anchorage in

Alaska is 13

4 in. What is the distance between these two towns to the nearest mile?

Practice Questions:

3,7,8,12,13,17 pg 11-13


Page 7

Section 1.3: Relating SI (Metric) and Imperial Units

Conversions Between Imperial and Metric System

Each measurement in the imperial system relates to a corresponding measurement in

the SI system.

The table below shows some approximate relationships between imperial units and SI

units:

Metric to Imperial Imperial to Metric

1 𝑚𝑚 =̇ 0.04 𝑖𝑛 1 𝑖𝑛 = 2.54𝑐𝑚 =̇ 2.5 𝑐𝑚

1 𝑐𝑚 =̇ 0.4 𝑖𝑛 1 𝑓𝑡 =̇ 30 𝑐𝑚 1 𝑓𝑡 =̇ 0.3 𝑚

1 𝑚 =̇ 39 𝑖𝑛 1 𝑚 =̇ 3.25 𝑓𝑡

1 𝑦𝑑 =̇ 90 𝑐𝑚 1 𝑦𝑑 =̇ 0.9 𝑚

1 𝑘𝑚 =̇ 0.6 𝑚𝑖 1 𝑚𝑖 =̇ 1.6 𝑘𝑚

Converting from Metres to Feet

Ex1. A bowling lane is approximately 19m long. What is this measurement to the nearest

foot?

Ex2. A Canadian football field is approximately 59m wide. What is this measurement to

the nearest foot?


Page 8

Converting Between Miles and Kilometres

Ex1. After meeting in Emerson, Manitoba, Hana drove 62 mi. south and Farrin drove

98 km north. Who drove farther?

Ex2. After meeting in Osoyoo, B.C., Takoda drove 114 km north and Winona drove 68

mi. south. Who drove farther?

Solving a Problem that Involves Unit Conversions

Ex1. Chris is 6ft. 2in. tall. To list his height on his driver’s license application, Chris

needs to convert this measurement to centimetres. What is Chris’ height to the nearest

centimetre?

Ex2. Nora knows that she is 5ft. 7in. tall. What height in centimetres will she list on her

driver’s license application?


Page 9

Estimating and Calculating Using Unit Conversions

Ex1. A truck driver knows that her semitrailer is 3.5 m high. The support beams of a

bridge are 11 ft. 9 in. high. Will the vehicle fit under the bridge? Justify the answer.

Ex2. A truck driver knows that his load is 15 ft. wide. Regulations along his route state

that any load over 4.3 m wide must have wide-load markers and an escort with flashing

lights. Does this vehicle need wide-load markers? Justify your answer.

Practice Problems:

4 – 13 pg 22-23


Page 10

Section 1.4: Surface Areas of Right Pyramids and Right Cones

Right Pyramids

A right pyramid is a 3-dimensional object that has triangular faces and a base that is a

polygon. The shape of the base determines the name of the pyramid.

Terminology:

Apex:

The point at which the triangular faces meet.

Slant Height:

The height of a triangular face of a right pyramid.

Height:

The height of the pyramid is the perpendicular distance from the apex to the

centre of the base.

Different types of triangular pyramids:

NOTE: A tetrahedron is a triangular pyramid. A regular tetrahedron has 4 congruent

equilateral triangular faces.

Apex

SlantHeight

Height

RectangularTetrahedron

Right Square Pyramid

Right Hexagonal

Pyramid


Page 11

Surface Area of Basic Shapes

Shape Formula Symbol Meanings Triangle

A𝑇𝑟𝑖𝑎𝑛𝑔𝑙𝑒 =1

2𝑏ℎ

b = base h = height

Rectangle

A𝑅𝑒𝑐𝑡𝑎𝑛𝑔𝑙𝑒 = 𝑙𝑤 l = length w = width

Circle

A𝐶𝑖𝑟𝑐𝑙𝑒 = 𝜋𝑟2 r = radius

(note: 𝑟 =1

2𝑑)

Right Polygon

A𝑃𝑜𝑙𝑦𝑔𝑜𝑛 =1

2𝑎𝑝

a = apothem p = perimeter

(note:p=(side length)(number of

sides))

Surface Area of a Regular Pyramid

Ex: Determine the area of the right square pyramid with a slant height of 10 cm and a

base side length of 8 cm.

h

b

l

w

r

a

To determine the surface area of a pyramid we simply determine the sum of the area of each surface.

10cm

8cm


Page 12

Determining the Surface Area of a Regular Pyramid Given its Slant Height

Ex1. Jeanne-Marie measured then recorded the lengths of the edges and slant height of

this regular tetrahedron. What is its surface area to the nearest square centimetre.

Ex2. Calculate the surface area of the regular tetrahedron to the nearest square metre.

9.0cm

7.8cm

5.0m

4.3m


Page 13

Determining the Surface Area of a Regular Pyramid

Ex1. A right rectangular pyramid has base dimensions 8ft by 10ft and a height of 16ft.

Calculate the surface area of the pyramid to the nearest square foot.

(hint: drawing a diagram may help)

Ex2. A right rectangular pyramid has base dimensions 4m by 6m and a height of 8m.

Calculate the surface area of the pyramid to the nearest square metre.

(hint: drawing a diagram may help)


Page 14

Ex3. Given that a regular pentagonal pyramid has a base side length of 12cm and

apothem of 7cm, determine the surface area if its height is 10cm. Give your answer to

the nearest square centimetre.

Ex4. Given that a regular pentagonal pyramid has a base side length of 1.8in and

apothem of 4.3inh, determine the surface area if its height is 3.7in. Give your answer to

the nearest square inch.


Page 15

Surface Area of a Right Cone

Ex1. A right cone has a base radius of 2ft and a height of 7ft. Calculate the surface area of

this cone to the nearest square foot.

Ex2. A right cone has a base diameter of 8m and a height of 10m. Calculate the surface

area of this cone to the nearest square foot.

𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 = 𝐿𝑎𝑡𝑒𝑟𝑎𝑙 𝐴𝑟𝑒𝑎 + 𝐵𝑎𝑠𝑒

For a right cone with slant height s and base radius r.

𝑆𝐴𝐶𝑜𝑛𝑒 = 𝜋𝑟𝑠 + 𝜋𝑟2

Where:

s = slant height

r = radius of base

s

r


Page 16

Terminology:

Lateral Area:

The area of the triangular faces of a pyramid or cone.

Determine an Unknown Measurement

Ex1. The lateral area of a cone is 220 cm2. The diameter of the cone is 10cm. Determine

the height of the cone to the nearest tenth of a centimetre.

Ex2. A model of the Great Pyramid of Giza is constructed for a museum display. The

surface area of the triangular faces is 3000 in2. The side length of the base is 50 in.

Determine the height of the model to a tenth of an inch.

Practice Problem

5, 7, 8, 9, 10, 13, 15, 16, & 18 pg 34-35


Page 17

Section 1.5: Volumes of Right Prisms, Pyramids, Cylinders, & Cones

Volume of a Prism

Ex1. Determine the area of the rectangular prism that has a base with dimensions 13 in

by 9.5 in and a height of 17 in.

Ex2. Determine the area of the 8.4 cm tall triangular prism that has a base with a length

of 2.8 cm and a base height of 1.7 in.

All Prisms, regardless of base shape, have the same formula for its Volume

𝑉𝑜𝑙𝑢𝑚𝑒 = (𝐴𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑎𝑠𝑒)(ℎ𝑒𝑖𝑔ℎ𝑡)

𝑉𝑃𝑟𝑖𝑠𝑚 = 𝐴𝑏𝑎𝑠𝑒ℎ h

A base


Page 18

Volume of a Pyramid

Ex1. Determine the volume of a rectangular pyramid with base dimensions of 12 in by

6 in with a height of 8 in.

Ex2. Determine the volume of a 15 mm tall hexagonal pyramid with apothem of 8 mm

and side length of 12 mm.

All pyramids, regardless of base shape, have the same formula for its Volume

𝑉𝑜𝑙𝑢𝑚𝑒 =1

3(𝐴𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑎𝑠𝑒)(ℎ𝑒𝑖𝑔ℎ𝑡)

𝑉𝑃𝑦𝑟𝑎𝑚𝑖𝑑 =1

3𝐴𝑏𝑎𝑠𝑒ℎ or 𝑉𝑝𝑦𝑟𝑎𝑚𝑖𝑑 =

1

3𝑉𝑝𝑟𝑖𝑠𝑚

A

h

b a s e


Page 19

Determining Volume of a Pyramid Given Slant Height

Ex1. Calculate the volume of a right square pyramid to the nearest cubic inch given that

it has a base length of 4 in and a slant height of 6 in.

Ex2. Calculate the volume of the right square pyramid to the nearest square foot given

that it has a height of 7 ft and a base length of 2 ft.


Page 20

Determining the Volume of a Cone or Cylinder

Ex1. Determine the volume of a cylinder with a diameter of 36 cm and a height of 42 cm.

Ex2. Determine the volume of a cylinder with a radius of 12.3 m and a height of 18 m.

Ex3. Determine the volume of the cone that has diameter of 36 cm and height of 42 cm.

Ex4. Determine the volume of the cone that has radius of 1.6 ft and height of 4.2 ft.


Page 21

Determining an Unknown Measurement

Ex1. A cone has a height of 4 yd and a volume of 205 yd3. Determine the radius to the

nearest yard.

Ex2. A cone has a height of 8 m and a volume of 300 m3. Determine the radius of the

base of the cone to the nearest metre.

Practice Problems

4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 18 pg 42-43


Page 22

Section 1.6: Surface Area and Volume of a Sphere

Surface Area of a Sphere

Ex1. The diameter of a baseball is approximately 3 in. Determine the surface area of a

baseball to the nearest square inch.

Ex2. The diameter of a softball is approximately 4 in. Determine the surface area of a

softball to the nearest square inch.

The surface area of a sphere with radius r is:

𝑆𝐴𝑆𝑝ℎ𝑒𝑟𝑒 = 4𝜋𝑟2 r


Page 23

Determining an Unknown Quantity Given Surface Area

Ex1. The surface area of a lacrosse ball is approximately 20 square inches. What is the

diameter of the lacrosse ball to the nearest tenth of an inch.

Ex2. The surface area of a soccer ball is approximately 250 square inches. What is the

diameter of a soccer ball to the nearest tenth of an inch.


Page 24

Volume of a Sphere

Ex1. The sun is approximately 870000 mi in diameter. What is the approximate volume

of the sun?

Ex2. The moon has an approximate diameter of 2160 mi. What is the approximate

volume of the moon?

The Volume of a sphere with radius r is:

𝑉𝑆𝑝ℎ𝑒𝑟𝑒 =4

3𝜋𝑟3

r


Page 25

Volume of Hemisphere

Surface Area of a Hemisphere

Ex1. A hemisphere has a radius of 8.0 cm.

(a) What is the volume of the hemisphere to the nearest tenth of a cubic centimetre?

(b) What is the surface area of the hemisphere to the nearest tenth of a square

centimetre (given that the top circular portion is included)?

The Volume of a hemisphere with radius r is:

𝑉𝐻𝑒𝑚𝑖𝑠𝑝ℎ𝑒𝑟𝑒 =1

2𝑉𝑆𝑝ℎ𝑒𝑟𝑒

The Surface Area of a hemisphere with radius r is:

𝑉𝐻𝑒𝑚𝑖𝑠𝑝ℎ𝑒𝑟𝑒 = 2𝜋𝑟2

This is only used when the circular portion is not used (only the rounded side)

The Volume of a sphere with radius r is:

𝑉𝐻𝑒𝑚𝑖𝑠𝑝ℎ𝑒𝑟𝑒 = 3𝜋𝑟2

This is only used when the circular portion is required

r

r

r


Page 26

Ex1. A hemisphere has a radius of 5.0 cm.

(a) What is the volume of the hemisphere to the nearest tenth of a cubic centimetre?

(b) What is the surface area of the hemisphere to the nearest tenth of a square

centimetre (given that the top circular portion is included)?

Practice Problems

3, 4, 7, 8, 9, 10, 11, 12 pg 51-52


Page 27

Section 1.7: Solving Problems Involving Objects

Determining the Volume of a Composite Object

Ex. Determine the volume of the composite object to the nearest tenth of a cubic unit.

(a)

(b)

2 .1 m

6 .7 m

2 .9 m

2 .9 m

32.0 cm

18.0 cm


Page 28

Determining Surface Area of a Composite Object

Ex. Determine the surface area of the composite object to the nearest tenth of a square

unit.

(a)

4 ft

2 ft


Page 29

(b)

6 m

6 m

5 m

4 m


Page 30

(c)

Practice Questions

#3,4(b),5,6,9,12 pg 59-61

8 m

6 in

5 m

4 in

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Chapter 1: Measurement - Mr. McDonald's Homepagemrmcdonaldshomepage.weebly.com/uploads/1/3/9/2/13921997/math_1201... · Chapter 1: Measurement Section 1.4 Page 12 Determining the