Unit 4: Circles and Area



Unit 4: Circles and Area – Grade 7 Name:__________________

Sec 4.1: Investigating Circles

Parts of a Circle

1. The distance from the center of the circle r

to any point on the circle is called the radius.

( symbol for radius is r )

* more than one radius is called radii.

2. A line passing through the center of a circle

with both endpoints on the circle is called the r r

diameter. ( symbol for diameter is d ). d

* Notice that one diameter equals two radii.

d = 2 × r or d = 2r

[ One full circle has 3600 – angles are measured using degrees. ]

Examples:

1. Find diameter

a). Answer: You are given the radius of the circle, so to

find diameter, multiply the radius by two.

4 cm d = 2r d = 4 ×2 d = 8 cm

b). Radius = 6 cm. Answer: d = 2r d = _______ d = _________

2. Find radius.

a). Answer: You are given the diameter of the circle, so to

10 cm find the radius, you must divide by 2.

r = 10 ÷ 2 r = 5 cm

b). Diameter = 15 cm Answer: r = 15 ÷ 2 r = ________

** NOTE **

If d = 2r then r = d ÷ 2 or r = d .

2

3a). A circle has a radius of 1.7 cm. What is the diameter of the circle?

Answer: if r = 1.7 then d = ________ d = _________

b). A circle has a diameter of 8.6 m. What is the radius of the circle?

Answer: if d = 8.6 then r = ________ r = _________

4. An artist has a sheet of poster paper measuring 1.6 m by 2.2 m, as shown.

2.2 m

What is the radius of the largest

1.6 m circle that can be cut from the paper?

Answer:

2.2 m

The largest radius would be 0.8 m

1.6 m ( consider 1.6 m the diameter of the

circle. A circle using 2.2 m as the

0.8 m diameter would not fit on the paper).

Practice: Worksheet 4.1 Investigating Circles

Constructing Circles

*** For this lesson students will need a compass, ruler and a string.

There are many ways to draw circles.

#1: Constructing a circle using a compass and a ruler.

a). Draw a circle with a radius of 2 cm.

► make a dot on your paper. This will be the center of your circle.

► using your compass and ruler, open your compass to 2 cm.

► place the pointy end of your compass on the dot, and draw your

circle. Be careful your compass doesn’t change size while drawing

your circle.

► when finished, measure the radius to see if you are correct.

b). Draw a circle with a diameter of 3 cm.

► find the radius first and follow the same steps as above.

r = 3 = 1.5 cm

2

#2: Constructing a circle with a string and a ruler.

► Tie a piece of string to the bottom of your pencil.

► Hold the string the length of the radius away from the pencil.

► Hold the string down against the paper where you want the center of the

circle to be.

► Draw around the center while keeping the string tight and the pencil upright.

#3: You can also trace circular objects.

Questions: Construct the following circles using any method of your choice.

a). with a radius of 1.5 cm b). with a diameter of 5 cm

c). Compare your circles with your classmates. What do you notice?

( They should be exactly the same… congruent ! )

Circle Activity

*** For this lesson students will need 3 circular objects, string, ruler and a calculator.

Grade 7 Math Circle Activity

Get in groups of 4 people. Name: ____________________ Name: __________________

Name: ____________________ Name: __________________

Each person was asked to bring 3 circular objects of different sizes. Place all objects in the center of your group for all group members to use. You will also need a string, a ruler and a calculator.

Directions:

* Record your information in the first four columns of the table below.

* Use the string to measure ( in centimeters ) the distance around 10 objects.

* Measure the radius and diameter of each object.

|Object |Distance Around the object ( |Radius |Diameter |Distance Around |

| |cm) |(cm) |(cm) |÷ Diameter |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

| | | | | |

Questions

1. Using your calculator, calculate the distance around ÷ diameter, for each object .

Record your answers in column 5 of the table.

2a). What do you notice about the answers in column 5?

2b). Does the size of the circle affect the answers?

3. Write an equation to find the distance around any circle.

Unit 4: Circles and Area – Grade 7 Name:__________________

Sec 4.2: Circumference of a Circle

The distance around the circle is called the circumference. It is the perimeter of the circle.

The circumference, C, of any circle be found using the equation C = π × d

C = π d

The symbol, π, is a Greek letter called “pi”. It represents the number 3.141592653589…

* this is known as an irrational number because it is a decimal that never ends and never

repeats. We estimate π = 3.14

Since the diameter = 2 × radius, we can also use the following equation to find circumference: C = π × 2r or C = 2 π r

Circumference, radius and diameter of a circle are all distance measurements. Therefore, the units are cm, mm, m, km, etc.

Examples:

1. A toonie has a radius of 1.4 cm

a). Find the diameter Answer: d = 2 × r

d = _______

d = _______

b). Find the Circumference

Answer: C = π × d or C = 2 π r

C = ___________ C = __________

C = ___________ C = ___________

2. A circular swimming pool has a circumference of 12 m. Find the diameter and the

radius of the pool. (Round answers to one decimal place).

Answer: If C = π × d then d = C = = d = _______

π

Radius = d = ___ = r = _______

2

3. Find the circumference.

Answer: Answer:

a). b).

15m

1.8 m

c). 2.2 m d). 11 m

More Practice with Circumference, Diameter, Radius.

Example 1: a). From the circle below:

► identify 3 radii ► the diameter

B

A C b). if OB = 6 cm,

O ► find AC ► the circumference

AC = C =

2a). If you double the diameter, what happens to the circumference of a circle?

Answer:

Try: d = 5 cm

Try: d = 10 cm

b). If you triple the diameter, what happens to the circumference of a circle?

Answer:

Try: d = 15 cm

3. If the circumference of a circle is 4.5 m, what is the radius in centimeters?

Answer:

The radius of the circle is _________.

Unit 4: Circles and Area – Grade 7 Name:__________________

Sec 4.3: Area of a Parallelogram

Area = the number of square units needed to cover a region (cm2, mm2, m2 )

EX: a mat covering an “area” on the floor.

Parallelogram = a quadrilateral ( 4-sided figure) where opposite sides are parallel.

Examples of Parallelograms

Square Rectangle Rhombus

Parallelogram NOT a parallelogram

because one pair of sides

Parallelogram is not parallel.

Area of a Parallelogram = base × height

A = b × h or A = bh

* The area of a parallelogram is the same as the area of a rectangle. WHY?

* A figure will keep the same area even if the orientation is changed or when it is broken

into smaller parts and rearranged. (It still has the same area - area is conserved)

Parallelogram Cut off Triangle

height

base

Replace Triangle to Now we have a Rectangle

create a Rectangle with the same base and height

as the original parallelogram

Therefore,

height a rectangle and

parallelogram

with the same base

and height will

have the same area

base

* the height and base of a parallelogram must meet at 900.

base

The height can be

drawn outside the

height parallelogram too,

height to ensure we have

900.

base

* Any side of the parallelogram can be the base. Then the height is the line that joins the

base to its parallel side. This line must be perpendicular ( 900 ) to the base.

* A variety of parallelograms with different shapes can have the same area.

Examples:

1. Find the area of each parallelogram.

1.5 m

a). Answer: b). Answer:

6 cm 8.3 m

4 cm

2. What is the area of a parallelogram that has a base of 9 m and a height of 3 m ?

Answer:

3. Find the missing dimension.

a). Answer:

A = 20 cm2 height ?

5 cm

b). Answer:

9.6 cm

A = 24 cm2

base ?

Unit 4: Circles and Area – Grade 7 Name:__________________

Sec 4.4: Area of a Triangle

Remember: Area of a Parallelogram = base × height

A = bh

height

base

If we draw a diagonal in So how can we find

the parallelogram we make the area of a triangle?

two congruent triangles

(same size and the same shape).

* Each triangle is 1 the area of the parallelogram Area of a Triangle

2 A = b×h

* As with parallelograms the base and height of the 2

triangle must meet at 900 and since its area the units

are still squared.

Examples: Find the area.

a). 17 cm A =

9cm

b). A =

5.6 m

4.2 m * the 5.6 m is not needed in this question.

( it is unnecessary information)

3.1 m

More Examples of Area of a Triangle

6cm

1. Daniel just bought a new sailboat with

sails that need to be replaced. B A

3cm 4cm

a). How much fabric will Daniel need to

replace sail A ?

b). How much fabric will Daniel need to

replace sail B?

2. Find the missing measurement.

Area = 9.9 m2

a). Area = 36 m2 b). height ? 5.5 m

8m

base ?

3. Sarah wants to make a triangular flag with a height of 0.8 m and a base of 1.2 m.

a). Find the amount of material she will need to make the flag.

b). If the material costs $5.95 per square meter, find the cost of the flag.

Unit 4: Circles and Area – Grade 7 Name:__________________

Sec 4.5: Area of a Circle

* Students will work through the Explore Activity on p.148. Continue with reading

through to p.150 on how to develop the formula for an area of a circle.

Remember π = 3.14

Area of a Circle = π × r × r and r = radius

Area = π r 2

* Since its area the units are still squared.

Examples:

1. Find the area of each circle. Round answers to the nearest tenth…one decimal place.

a). Area = π × r × r

5 cm

b). diameter = Area = π × r × r

8.4 m

radius =

2. The diameter of a dime is 1.8 cm. Find the area of the dime.

diameter = Area = π × r × r

radius =

1.8 cm

More Circle Examples

1. A pizza has a diameter of 30 cm. diameter =

30 cm

a). Find the area of the pizza.

radius =

Area = π × r × r

b). If the pizza is split equally Area of each slice =

among 6 friends, what is the

area of each slice? Area of each slice =

2. Find the area of the following donut.

*** Hint: Find the area of the big outer circle and then

subtract the area of the small inner circle and

3 cm and that will give you the area of the donut.

8 cm

Area of the Big Circle

= π × r × r

8 cm

Area of the Small Circle

= π × r × r

3 cm

Area of the Donut = Area of Big Circle – Area of the Small Circle

3. The circumference of a circle is 50.24 cm.

a). Find the diameter of the circle.

b). Find the radius of the circle.

c). Find the area of the circle.

Unit 4: Circles and Area – Grade 7 Name:__________________

Sec 4.5: Interpreting Circle Graphs

Circle graph – is a diagram that uses parts of a circle to display data.

Each sector of a circle graph represents a percent of

the whole circle.

sector

All sectors must add up to 100 % because the whole circle is 100 %.

Every Circle Graph should include:

► a title

► each sector labeled with a category and percent

( some circle graphs have a legend on the side to show the categories).

Example

1. 60 grade 7 students were surveyed about their favorite after-school activity. The

results are shown in the circle graph below.

After School Activities

20%

Play

30% Hang Sports

Out with 10% Watch TV

Friends

5% Read

25% 10%

Play Video Games Do Homework

a). Which activity is the most popular?

is the least popular?

is the favorite of 1 of the students ?

4

b). How many students ….. hang out with friends?

30 % of 60

….. do their homework?

10 % of 60

….. play sports?

….. watch TV?

….. play video games?

….. read ?

A way to check your answer is to add up all the students – it should equal 60.

Unit 4: Circles and Area – Grade 7 Name:__________________

Sec 4.5: Drawing Circle Graphs

A circle graph show different sectors of a circle (smaller parts of the whole circle). Each sector has an angle measurement in degrees. We use a protractor to draw these angles. These angles are called central angles.

How many degrees in a full circle?

360 0

* Therefore, the sum of all central

central angle angles is 360 0

Use your protractor, follow the steps below and draw the following angles:

Step #1: draw a dot.

Step #2: from the dot, draw a horizontal starting line

Step #3: place the center of your protractor on the dot and line up the horizontal line with

0 on your protractor.

a). 36 0 b). 54 0

( * please note none of these measurements are not accurate but an approximation of

what your angles should look like using your protractor. )

c). 68 0 d). 94 0

* Notice….what is the sum of these angles?

e). 108 0 360 0

* How can we draw a circle graph

containing all of these angles?

Steps to Drawing a Circle Graph

#1: Draw a circle. Place a dot at the center and a starting line (a radius).

#2: Using your protractor, measure the first angle (start at 0 0 ).

36 0

#3: To draw your next angle, begin where you left off from the first angle drawn. You

now have a new starting line.

* Continue step 3 until all angles are drawn

#4: Label each sector, include percent and title your circle graph.

* Hint : It’s best to draw the largest angle last.

68 0 54 0

36 0

94 0

108 0

Examples

1. Using a protractor and a ruler construct and identify the following angles.

a). 300 b). 110 0 c). 180 0 d). 220 0 e). 900

|a). |b). |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

|This is called an acute angle. The measure is less than 900. |This is called an obtuse angle. The measure is greater than 900 but less than |

| |1800. |

|c). |d). |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

|This is called a straight angle. The measure is exactly 1800. It is a straight |* This is known as a reflex angle. It’s size is greater than 1800 but less |

|line. |than 3600. To construct this angle subtract it’s measurement from 3600 and |

| |make the smaller angle. (360 – 220 = 140).The larger angle is the reflex angle.|

| | |

|e). | |

| | |

| | |

| | |

| | |

| | |

| | |

|* This angle is called a right angle. It is exactly 900. | |

|A 900 angle is usually identified by . | |

2. The circles below are divided into three sections. The central angles of two sections

are given. Find the measure of the third central angle.

a). 550 , 1170 Answer:

b). 910 , 740 Answer:

c). 10 , 630 Answer:

3. Cory has five angle measurements 350 , 800 , 600 , 1350 and 500 . Can he draw a circle

graph with these central angles.

Answer:

Drawing Circle Graphs When Not Given Angle Measurements.

Examples: Draw a circle graph for each situation.

1. A survey was conducted by the school council for the question “ Should Munden

Drive be a one-way street?” The answers are recorded in the table below.

a). Complete the table and construct a circle graph.

|Answers |Percent |Decimal |Degrees |

| | |( divide percent by 100 ) |( Decimal × 360 ) |

|Yes |43% | | |

|No |35% | | |

|Undecided |12% | | |

|No Opinion |10% | | |

Check: 100 % 1.00 3600

* Always check each column. The percents should add to be 100. The decimals should add to be 1 and the degrees should add to be 3600. Often we have to round the angle measurements because the degrees doesn’t work out exactly. So if the degree column adds to be 359 or 361 that’s okay.

Should Munden Drive be a One-Way Street?

No

35 %

12%

Undecided

10%

No Opinion

Yes

43%

• It doesn’t matter what order you draw your angles in. It’s just a good idea to save the biggest for last…especially if it measures more than 1800.

2. Two grade 7 classes were asked how they get to school each day. The results are:

9 rode their bikes, 11 walked. 17 got the bus and 13 by car. Construct a circle graph

to display this data.

• Construct and complete a table first, it will help you draw your circle graph.

|Way to School |# of Students |Fraction of Total |Percent of Total |Percent as a Decimal |Degrees of Circles |

|Bike |9 | | | | |

| | | | | | |

|Walk |11 | | | | |

| | | | | | |

|Bus |17 | | | | |

| | | | | | |

|Car |13 | | | | |

| | | | | | |

|Total |50 | | | | |

| | | | | | |

* In this example you needed to find the total first – which was 50. Then set up the fractions using 50 and change to fractions out of 100. We need fractions out of 100 because that’s our percent and then we use percent to find degrees.

How Students Get to School

Walk

22%

Bike

18%

Bus

34%

Car

26%

-----------------------

You should always ask yourself “Does that answer make sense?” If you estimate 3.14 × 2.8 to 3 × 3, then we know the answer should be approximately 9. So 8.8 is a reasonable answer.

estimate

___ =

estimate

__ =

Since 1 m = 100 cm,

then 0.7 m = 70 cm

Just like × by 100. Move the decimal two places to the right.

6 cm

15 cm

To complete on loose leaf: p.132 #3,4 and 6.

To complete on loose leaf: p.131& 132 #1 & 2.

To complete on loose leaf: p.136 # 1, 2, 4.

To complete on loose leaf: p.141& 142 # 2, 6, 8, 10.

To complete on loose leaf: p.145 & 146 #2 and 4.

To complete on loose leaf: p.146 – 147 #5, 9.

To complete on loose leaf: p.151 #1 and 2

To complete on loose leaf: p.151 #5 and 7.

To complete on loose leaf: p.158 # 1and 2.

To complete on loose leaf: p.163& 164 #1,2 and 4.

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