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UNIT 4 – CIRCLES & AREA

Section 4.1 – Investigating Circles

Circle: all the points in a plane that are the same distance away from a given

centre point

Symbols r – radius d – diameter C - circumference

Practice Problems

Identify and label the radius and diameter of each circle below.

A) B) C)

Radius: the distance

from the centre to

any point on the

circle. (plural radii)

Diameter: the longest

line segment touching

the circle at two

opposite points and

passing through the

centre.

Diameter is twice the

radius.

Circumference: the

distance around a

circle. A circle’s

perimeter.

The diameter of a circle is twice the

radius.

We can write,

Diameter = 2 x radius

Or,

d = 2r

The radius of a circle is half the

diameter

We can write,

Radius = 2

Diameter

Or

2

dr

3 cm 12 cm

2.5 cm

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Section 4.2 – Circumference of a Circle

Circumference: the distance around a circle (also called perimeter)

You can estimate the circumference and diameter of a circle using this

relationship.

dC 3 AND 3

Cd

Example Problems

A) Estimate the circumference of the circle below

B) Estimate the circumference of the circle below

When you divide the exact circumference of any circle by its exact diameter

the actual answer will always come out to be the same number for any circle.

That number is known as pi and is represented by the symbol

Pi ( ): The number you get when you divide the circumference of any

circle by its diameter.

Pi is known as an irrational number because it goes on forever and never

repeats in a pattern.

Irrational Number: a number that never ends and never repeats in a

pattern.

d = 5 cm

r = 2 cm

3

begins with 3. and is followed by a never ending series of digits

Here are the first 100 digits of pi after the decimal

3.1415926535 8979323846 2643383279 5028841971 6939937510

5820974944 5923078164 0628620899 8628034825 3421170679...

So to find the circumference and diameter of a circle more accurately we

should use instead of 3.

FORMULA for CIRCUMFERENCE of a circle dC

FORMULA for DIAMETER of a circle

Cd

Because pi never ends we usually round it to 3.14

~ 3.14

Or, we can use the button on our calculator to get the exact value.

PRACTICE PROBLEMS

1. A circle has diameter 10.5 cm.

Calculate the circumference of the circle to the nearest millimetre.

2. A circle has radius 4.3 mm.

Calculate the circumference of the circle to the nearest millimetre.

3. A circle has circumference 12.6 m.

Calculate the diameter of the circle to the nearest centimetre.

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Section 4.3 – Area of a Parallelogram

Area: the amount of flat, 2-dimensional space a shape covers.

Area is 2-dimensional, therefore it is measure in squared units (for

measuring two dimensional amounts).

These may include km2, hm2, dam2, m2, dm2 cm2, mm2

Area of a Parallelogram

Parallelogram: a four side figure having made of two sets of parallel lines.

Parallel Lines: Lines that are always an equal distance apart and therefore

never intersect each other.

Parts of a Parallelogram

NOTE: An equal

number of arrow heads

indicate that the lines

they are on are parallel

to each other.

Base (b): any side of

a parallelogram.

Height (h): the length of a

perpendicular line segment

joining two parallel sides.

If the parallelogram were rotated the

base and height would be as shown. h

b

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Determining the Formula for Area of a Parallelogram

If we were to cut any parallelogram along its height and reconnect it by

joining one set of parallel sides we would create a rectangle.

Notice that the length and width of the rectangle made correspond to the

height and base of the parallelogram we started with. This means to find

the area of the parallelogram we simply multiply its base by its height.

Area parallelogram = BASE X HEIGHT

A par = b X h

A par = bh

Example – What is the area of

the following parallelogram shown?

Section 4.4 – Area of a Triangle

Triangle: a 3 sided shape with 3 enclosed angles adding up to 180º

Parts of a Triangle

h

b b

h

5 mm

3 mm

A par = bh A par = 5 mm X 3 mm

A par = 15 mm2

base

height

Base (b): any side of a triangle

Height (h): the length of a

perpendicular line connecting the base

to the opposite vertex.

NOTE: Each point created by an angle

on a triangle is called a vertex. More

than one are called vertices.

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Determining the formula for Area of a Triangle

We can make two congruent triangles from any parallelogram just by cutting

along a diagonal.

Area triangle = heightbase2

1

A tri = bh2

1

or

A tri = 2

bh

PRACTICE PROBLEMS

1. The area of each triangle is given. Find each unknown measure.

a)

b)

h

b

h

b

b

h

Notice that the area of each

triangle is half of the area of

the parallelogram we started

with.

So, to get the area of a

triangle we simply take half of

the area of its related

parallelogram.

Ex. Find the area of the triangle below.

5 mm

9 mm

A tri = bh2

1

A tri = 592

1 =

2

45 = 22.5 mm2

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c)

d)

Section 4.5 – Area of a Circle

Recall, diameternceCircumfere

Or radiusnceCircumfere 2

We can use this knowledge to develop a formula for finding the area of a

circle. Take a circle divided into equal sectors.

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Notice that the height of the parallelogram is the same as the radius of the

circle.

The base of the parallelogram is the same as half the circle’s circumference

Area parallelogram = BASE X HEIGHT

Area parallelogram = rr

2

2

Area parallelogram = rr

2

2 122

Area parallelogram = rr

Area parallelogram = 2r

Area circle = 2r

PRACTICE PROBLEMS

1. Calculate the area of each circle.

Give the answers to one decimal place.

a) b) c)

1st -If we cut out each sector and arrange in a

shape resembling a parallelogram we get

something similar to the picture shown below

So to find the area of a circle we

use the formula

Area circle = 2r

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2. A carpenter is making a circular tabletop with radius 0.5 m.

What is the area of the tabletop to the nearest tenth of a metre?

3. The diameter of a knob on a CD player is 0.78 cm.

a) What is the radius of the knob?

b) What is the circumference of the knob?

c) What is the area of the knob?

Section 4.6 – Interpreting Circle Graphs

Circle Graphs – compares amounts by using area sectors of a circle.

Data is shown as a fraction (or percent) of a circles whole area. The circle

represents one whole or 100% of a set of data.

Each sector of a circle graph represents a percent of the whole.

Example: The following circle graph was produced using the data shown in

the table below.

Favorite Sports of Grade 7 Students at Greendale Jr. High

Sport Number of Students

Basketball 12

Baseball 7

Hockey 9

Tennis 2

Golf 1

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Favorite Sports of Grade 7 Students at Greendale

Jr.High

basketball

39%

baseball

23%

hockey

29%

tennis

6%

golf

3%

basketball

baseball

hockey

tennis

golf

In order to understand what a circle graph is displaying we need to be able

to analyze the data being presented.

Ex. 1. The circle graph shows Samson’s household budget for a month.

Each fraction of

the circle if

called a SECTOR.

The box showing what category

each sector represents is called

the LEGEND

a) Samson takes home $2500 per month. How much does he budget for each item?

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Ex 2. This circle graph shows how much time is spent in one day watching different types of TV programs.

c) Estimate the fraction of time spent watching sitcoms.

d) Suppose TV is watched for 1000 days.

Estimate how much time is spent watching sitcoms.

Section 4.7 – Drawing Circle Graphs

A circle’s angles add up to 360º

When a circle is divided into 100 equal parts

each sector is 1%

To draw circle graphs you need to be able to use a protractor to draw angles

of a certain measure.

You will also need to be able to divide a circle into sectors with given angles.

You may also be required to find missing angle measures based on known

measures.

a) Which type of program is watched for the

greatest amount of time?

b) Which two types of programs are watched for approximately

the same amount of time?

60 º

120 º

90 º

90 º

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For each circle below, find the missing angle measures.

You will also need to be able to determine how many degrees a sector of a

circle takes up if given a percentage.

For example, in the circle graph below determine the measurement of each

angle in degrees.

117 º

55 º

91 º

74 º

45% 25%

30%

Since there are 360º in a circle we need

to find each percent of 360.

45% of 360º

0.45 X 360 = 162º

So, 45% of the circle is 162º

25% of 360º

30% of 360º

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To construct a circle graph from a data table there a three basic steps:

STEP 1 – Convert the data to a fraction of the whole data set

STEP 2 – Convert the fraction to a decimal

STEP 3 – Find the % as an angle (% x 360º)

Example:

A survey was conducted among 27 Grade 9 students at Greendale Jr. High to

find out students favorite kind of music. The results are shown below.

Follow the three steps to construct a circle graph for this data.

NOTE: Frequency means the number of data points (in this case votes) for a

category.

Type of Music Frequency

(27 total)

Fraction of

Total

Fraction as a

%

% as an angle

(% of 360)

Classical 2 2/27 0.07 = 7% 0.07 X 360 = 25.2º

Rap/Hip Hop 7

Rock 11

Country 4

Jazz/Blues 3

TITLE: __________________________________________________

LEGEND