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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