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Radian measure.
Radian measure. 1 / 12
Measuring angles
When someone talks about an angle of 1◦ how do you know what theymean?
You start by remembering that 360◦ corresponds to a full revolution.
So 1◦ should be one 360’th of a full revolution.
This is 1◦.
Zoomed in
Does this seem arbitrary to you? Why is 360◦ a full revolution?
Radian measure. 2 / 12
Measuring angles
When someone talks about an angle of 1◦ how do you know what theymean?You start by remembering that 360◦ corresponds to a full revolution.
So 1◦ should be one 360’th of a full revolution.
This is 1◦.
Zoomed in
Does this seem arbitrary to you? Why is 360◦ a full revolution?
Radian measure. 2 / 12
Measuring angles
When someone talks about an angle of 1◦ how do you know what theymean?You start by remembering that 360◦ corresponds to a full revolution.
So 1◦ should be one 360’th of a full revolution.
This is 1◦.
Zoomed in
Does this seem arbitrary to you? Why is 360◦ a full revolution?
Radian measure. 2 / 12
Measuring angles
When someone talks about an angle of 1◦ how do you know what theymean?You start by remembering that 360◦ corresponds to a full revolution.
So 1◦ should be one 360’th of a full revolution.
This is 1◦.
Zoomed in
Does this seem arbitrary to you? Why is 360◦ a full revolution?
Radian measure. 2 / 12
Measuring angles
When someone talks about an angle of 1◦ how do you know what theymean?You start by remembering that 360◦ corresponds to a full revolution.
So 1◦ should be one 360’th of a full revolution.
This is 1◦. Zoomed in
Does this seem arbitrary to you? Why is 360◦ a full revolution?
Radian measure. 2 / 12
Measuring angles
When someone talks about an angle of 1◦ how do you know what theymean?You start by remembering that 360◦ corresponds to a full revolution.
So 1◦ should be one 360’th of a full revolution.
This is 1◦. Zoomed in
Does this seem arbitrary to you? Why is 360◦ a full revolution?
Radian measure. 2 / 12
Radian measureToday’s goal is to develop a measure of angles which is less arbitrary.Imagine you are standing on a circle ofradius 1 foot.
Start walking counter-clockwise. Stopwhen you have covered s feet.You have now covered an angle of s radi-ans
Distance=s.s radians
Definition
An arc of length s on a circle of radius r covers an angle of sr radians.
Example: What are the radian measures of these angles?
2 cm
2 cm
1.5 cm
6 cm
Radian measure. 3 / 12
Radian measureToday’s goal is to develop a measure of angles which is less arbitrary.Imagine you are standing on a circle ofradius 1 foot.Start walking counter-clockwise. Stopwhen you have covered s feet.
You have now covered an angle of s radi-ans
Distance=s.s radians
Definition
An arc of length s on a circle of radius r covers an angle of sr radians.
Example: What are the radian measures of these angles?
2 cm
2 cm
1.5 cm
6 cm
Radian measure. 3 / 12
Radian measureToday’s goal is to develop a measure of angles which is less arbitrary.Imagine you are standing on a circle ofradius 1 foot.Start walking counter-clockwise. Stopwhen you have covered s feet.
You have now covered an angle of s radi-ans
Distance=s.
s radians
Definition
An arc of length s on a circle of radius r covers an angle of sr radians.
Example: What are the radian measures of these angles?
2 cm
2 cm
1.5 cm
6 cm
Radian measure. 3 / 12
Radian measureToday’s goal is to develop a measure of angles which is less arbitrary.Imagine you are standing on a circle ofradius 1 foot.Start walking counter-clockwise. Stopwhen you have covered s feet.You have now covered an angle of s radi-ans
Distance=s.
s radians
Definition
An arc of length s on a circle of radius r covers an angle of sr radians.
Example: What are the radian measures of these angles?
2 cm
2 cm
1.5 cm
6 cm
Radian measure. 3 / 12
Radian measureToday’s goal is to develop a measure of angles which is less arbitrary.Imagine you are standing on a circle ofradius 1 foot.Start walking counter-clockwise. Stopwhen you have covered s feet.You have now covered an angle of s radi-ans
Distance=s.s radians
Definition
An arc of length s on a circle of radius r covers an angle of sr radians.
Example: What are the radian measures of these angles?
2 cm
2 cm
1.5 cm
6 cm
Radian measure. 3 / 12
Radian measureToday’s goal is to develop a measure of angles which is less arbitrary.Imagine you are standing on a circle ofradius 1 foot.Start walking counter-clockwise. Stopwhen you have covered s feet.You have now covered an angle of s radi-ans
Distance=s.s radians
Definition
An arc of length s on a circle of radius r covers an angle of sr radians.
Example: What are the radian measures of these angles?
2 cm
2 cm
1.5 cm
6 cm
Radian measure. 3 / 12
Radian measureToday’s goal is to develop a measure of angles which is less arbitrary.Imagine you are standing on a circle ofradius 1 foot.Start walking counter-clockwise. Stopwhen you have covered s feet.You have now covered an angle of s radi-ans
Distance=s.s radians
Definition
An arc of length s on a circle of radius r covers an angle of sr radians.
Example: What are the radian measures of these angles?
2 cm
2 cm
1.5 cm
6 cm
Radian measure. 3 / 12
Converting between radians and degrees
Definition
An arc of length s on a circle of radius r covers an angle ofs
rradians.
If you know the degree measure of an angle, can you find its radianmeasure?Let’s start with 360◦.
Drawing a 360◦ arc on a circle of radius r is the same asdrawing a whole circle.If the radius is r , then what is the length of this arc? In otherwords, What is the circumference of the circle of radius r?s = 2πr .
This angle measures2πr
r=
2π
radians.
360◦ is the same as 2π radians.
Radian measure. 4 / 12
Converting between radians and degrees
Definition
An arc of length s on a circle of radius r covers an angle ofs
rradians.
If you know the degree measure of an angle, can you find its radianmeasure?Let’s start with 360◦.Drawing a 360◦ arc on a circle of radius r is the same asdrawing a whole circle.
If the radius is r , then what is the length of this arc? In otherwords, What is the circumference of the circle of radius r?s = 2πr .
This angle measures2πr
r=
2π
radians.
360◦ is the same as 2π radians.
Radian measure. 4 / 12
Converting between radians and degrees
Definition
An arc of length s on a circle of radius r covers an angle ofs
rradians.
If you know the degree measure of an angle, can you find its radianmeasure?Let’s start with 360◦.Drawing a 360◦ arc on a circle of radius r is the same asdrawing a whole circle.If the radius is r , then what is the length of this arc? In otherwords, What is the circumference of the circle of radius r?
s = 2πr .
This angle measures2πr
r=
2π
radians.
360◦ is the same as 2π radians.
Radian measure. 4 / 12
Converting between radians and degrees
Definition
An arc of length s on a circle of radius r covers an angle ofs
rradians.
If you know the degree measure of an angle, can you find its radianmeasure?Let’s start with 360◦.Drawing a 360◦ arc on a circle of radius r is the same asdrawing a whole circle.If the radius is r , then what is the length of this arc? In otherwords, What is the circumference of the circle of radius r?s = 2πr .
This angle measures2πr
r=
2π
radians.
360◦ is the same as 2π radians.
Radian measure. 4 / 12
Converting between radians and degrees
Definition
An arc of length s on a circle of radius r covers an angle ofs
rradians.
If you know the degree measure of an angle, can you find its radianmeasure?Let’s start with 360◦.Drawing a 360◦ arc on a circle of radius r is the same asdrawing a whole circle.If the radius is r , then what is the length of this arc? In otherwords, What is the circumference of the circle of radius r?s = 2πr .
This angle measures2πr
r=
2π
radians.
360◦ is the same as 2π radians.
Radian measure. 4 / 12
Converting between radians and degrees
Definition
An arc of length s on a circle of radius r covers an angle ofs
rradians.
If you know the degree measure of an angle, can you find its radianmeasure?Let’s start with 360◦.Drawing a 360◦ arc on a circle of radius r is the same asdrawing a whole circle.If the radius is r , then what is the length of this arc? In otherwords, What is the circumference of the circle of radius r?s = 2πr .
This angle measures2πr
r= 2π radians.
360◦ is the same as 2π radians.
Radian measure. 4 / 12
Converting between radians and degrees
Definition
An arc of length s on a circle of radius r covers an angle ofs
rradians.
If you know the degree measure of an angle, can you find its radianmeasure?Let’s start with 360◦.Drawing a 360◦ arc on a circle of radius r is the same asdrawing a whole circle.If the radius is r , then what is the length of this arc? In otherwords, What is the circumference of the circle of radius r?s = 2πr .
This angle measures2πr
r= 2π radians.
360◦ is the same as 2π radians.
Radian measure. 4 / 12
Converting between radians and degrees
So far: Since a 360◦ arc on a radius r circle has length = 2πr , 360◦ isequal to 2π radians.What about a 1◦ arc?
How many of these does it take to fill a 360◦ arc? a silly question.360 of them.So 360 · (The arc length of a 1◦ arc) = 2πr .What is the arc length of a 1◦ arc?2πr/360 = πr/180.
So the radian measure of a 1◦ arc isarc length
radius=πr/180
r=
π
180.
1◦ =π
180radians.
What about d◦
A d◦ angle is made of d angles of measure 1◦. These will total d · π
180radians.Concept check: Use this to compute radian measure of a 30◦ angle.
Radian measure. 5 / 12
Converting between radians and degrees
So far: Since a 360◦ arc on a radius r circle has length = 2πr , 360◦ isequal to 2π radians.What about a 1◦ arc?How many of these does it take to fill a 360◦ arc? a silly question.
360 of them.So 360 · (The arc length of a 1◦ arc) = 2πr .What is the arc length of a 1◦ arc?2πr/360 = πr/180.
So the radian measure of a 1◦ arc isarc length
radius=πr/180
r=
π
180.
1◦ =π
180radians.
What about d◦
A d◦ angle is made of d angles of measure 1◦. These will total d · π
180radians.Concept check: Use this to compute radian measure of a 30◦ angle.
Radian measure. 5 / 12
Converting between radians and degrees
So far: Since a 360◦ arc on a radius r circle has length = 2πr , 360◦ isequal to 2π radians.What about a 1◦ arc?How many of these does it take to fill a 360◦ arc? a silly question.360 of them.
So 360 · (The arc length of a 1◦ arc) = 2πr .What is the arc length of a 1◦ arc?2πr/360 = πr/180.
So the radian measure of a 1◦ arc isarc length
radius=πr/180
r=
π
180.
1◦ =π
180radians.
What about d◦
A d◦ angle is made of d angles of measure 1◦. These will total d · π
180radians.Concept check: Use this to compute radian measure of a 30◦ angle.
Radian measure. 5 / 12
Converting between radians and degrees
So far: Since a 360◦ arc on a radius r circle has length = 2πr , 360◦ isequal to 2π radians.What about a 1◦ arc?How many of these does it take to fill a 360◦ arc? a silly question.360 of them.So 360 · (The arc length of a 1◦ arc) = 2πr .What is the arc length of a 1◦ arc?
2πr/360 = πr/180.
So the radian measure of a 1◦ arc isarc length
radius=πr/180
r=
π
180.
1◦ =π
180radians.
What about d◦
A d◦ angle is made of d angles of measure 1◦. These will total d · π
180radians.Concept check: Use this to compute radian measure of a 30◦ angle.
Radian measure. 5 / 12
Converting between radians and degrees
So far: Since a 360◦ arc on a radius r circle has length = 2πr , 360◦ isequal to 2π radians.What about a 1◦ arc?How many of these does it take to fill a 360◦ arc? a silly question.360 of them.So 360 · (The arc length of a 1◦ arc) = 2πr .What is the arc length of a 1◦ arc?2πr/360 = πr/180.
So the radian measure of a 1◦ arc isarc length
radius=πr/180
r=
π
180.
1◦ =π
180radians.
What about d◦
A d◦ angle is made of d angles of measure 1◦. These will total d · π
180radians.Concept check: Use this to compute radian measure of a 30◦ angle.
Radian measure. 5 / 12
Converting between radians and degrees
So far: Since a 360◦ arc on a radius r circle has length = 2πr , 360◦ isequal to 2π radians.What about a 1◦ arc?How many of these does it take to fill a 360◦ arc? a silly question.360 of them.So 360 · (The arc length of a 1◦ arc) = 2πr .What is the arc length of a 1◦ arc?2πr/360 = πr/180.
So the radian measure of a 1◦ arc is
arc length
radius=πr/180
r=
π
180.
1◦ =π
180radians.
What about d◦
A d◦ angle is made of d angles of measure 1◦. These will total d · π
180radians.Concept check: Use this to compute radian measure of a 30◦ angle.
Radian measure. 5 / 12
Converting between radians and degrees
So far: Since a 360◦ arc on a radius r circle has length = 2πr , 360◦ isequal to 2π radians.What about a 1◦ arc?How many of these does it take to fill a 360◦ arc? a silly question.360 of them.So 360 · (The arc length of a 1◦ arc) = 2πr .What is the arc length of a 1◦ arc?2πr/360 = πr/180.
So the radian measure of a 1◦ arc isarc length
radius=
πr/180
r=
π
180.
1◦ =π
180radians.
What about d◦
A d◦ angle is made of d angles of measure 1◦. These will total d · π
180radians.Concept check: Use this to compute radian measure of a 30◦ angle.
Radian measure. 5 / 12
Converting between radians and degrees
So far: Since a 360◦ arc on a radius r circle has length = 2πr , 360◦ isequal to 2π radians.What about a 1◦ arc?How many of these does it take to fill a 360◦ arc? a silly question.360 of them.So 360 · (The arc length of a 1◦ arc) = 2πr .What is the arc length of a 1◦ arc?2πr/360 = πr/180.
So the radian measure of a 1◦ arc isarc length
radius=πr/180
r=
π
180.
1◦ =π
180radians.
What about d◦
A d◦ angle is made of d angles of measure 1◦. These will total d · π
180radians.Concept check: Use this to compute radian measure of a 30◦ angle.
Radian measure. 5 / 12
Converting between radians and degrees
So far: Since a 360◦ arc on a radius r circle has length = 2πr , 360◦ isequal to 2π radians.What about a 1◦ arc?How many of these does it take to fill a 360◦ arc? a silly question.360 of them.So 360 · (The arc length of a 1◦ arc) = 2πr .What is the arc length of a 1◦ arc?2πr/360 = πr/180.
So the radian measure of a 1◦ arc isarc length
radius=πr/180
r=
π
180.
1◦ =π
180radians.
What about d◦
A d◦ angle is made of d angles of measure 1◦. These will total d · π
180radians.Concept check: Use this to compute radian measure of a 30◦ angle.
Radian measure. 5 / 12
Converting between radians and degrees
So far: Since a 360◦ arc on a radius r circle has length = 2πr , 360◦ isequal to 2π radians.What about a 1◦ arc?How many of these does it take to fill a 360◦ arc? a silly question.360 of them.So 360 · (The arc length of a 1◦ arc) = 2πr .What is the arc length of a 1◦ arc?2πr/360 = πr/180.
So the radian measure of a 1◦ arc isarc length
radius=πr/180
r=
π
180.
1◦ =π
180radians.
What about d◦
A d◦ angle is made of d angles of measure 1◦. These will total d · π
180radians.Concept check: Use this to compute radian measure of a 30◦ angle.
Radian measure. 5 / 12
Converting between radians and degrees
So far: Since a 360◦ arc on a radius r circle has length = 2πr , 360◦ isequal to 2π radians.What about a 1◦ arc?How many of these does it take to fill a 360◦ arc? a silly question.360 of them.So 360 · (The arc length of a 1◦ arc) = 2πr .What is the arc length of a 1◦ arc?2πr/360 = πr/180.
So the radian measure of a 1◦ arc isarc length
radius=πr/180
r=
π
180.
1◦ =π
180radians.
What about d◦
A d◦ angle is made of d angles of measure 1◦. These will total d · π
180radians.Concept check: Use this to compute radian measure of a 30◦ angle.
Radian measure. 5 / 12
Converting between radians and degrees
So far: Since a 360◦ arc on a radius r circle has length = 2πr , 360◦ isequal to 2π radians.What about a 1◦ arc?How many of these does it take to fill a 360◦ arc? a silly question.360 of them.So 360 · (The arc length of a 1◦ arc) = 2πr .What is the arc length of a 1◦ arc?2πr/360 = πr/180.
So the radian measure of a 1◦ arc isarc length
radius=πr/180
r=
π
180.
1◦ =π
180radians.
What about d◦
A d◦ angle is made of d angles of measure 1◦. These will total d · π
180radians.Concept check: Use this to compute radian measure of a 30◦ angle.
Radian measure. 5 / 12
Converting between radians and degrees
Theorem
To convert from degrees to radians, multiply by π180 .
To convert from radians to degrees multiply by 180π .
Here is a right angle, which measures 90◦.What is its radian measure?
Radian measure. 6 / 12
Arc length and sector area
Suppose you have a blueberry pie or radius rand you cut out a piece whose angle is π/4radians.Let’s find out how much area this piece of piehas.
How many of these pieces are in the whole ofthe pie? How many times does π
4 fit into2π? 2π
π/4 =
8
There are eight pieces like this in the wholepie, whose total area is:
πr2
.
The area of this piece iswhole area
8=
πr2
8
Next slide: What if we cut a different angle? Then how much area do wehave?
Radian measure. 7 / 12
Arc length and sector area
Suppose you have a blueberry pie or radius rand you cut out a piece whose angle is π/4radians.Let’s find out how much area this piece of piehas.
How many of these pieces are in the whole ofthe pie?
How many times does π4 fit into
2π? 2ππ/4 =
8
There are eight pieces like this in the wholepie, whose total area is:
πr2
.
The area of this piece iswhole area
8=
πr2
8
Next slide: What if we cut a different angle? Then how much area do wehave?
Radian measure. 7 / 12
Arc length and sector area
Suppose you have a blueberry pie or radius rand you cut out a piece whose angle is π/4radians.Let’s find out how much area this piece of piehas.
How many of these pieces are in the whole ofthe pie? How many times does π
4 fit into2π?
2ππ/4 =
8
There are eight pieces like this in the wholepie, whose total area is:
πr2
.
The area of this piece iswhole area
8=
πr2
8
Next slide: What if we cut a different angle? Then how much area do wehave?
Radian measure. 7 / 12
Arc length and sector area
Suppose you have a blueberry pie or radius rand you cut out a piece whose angle is π/4radians.Let’s find out how much area this piece of piehas.
How many of these pieces are in the whole ofthe pie? How many times does π
4 fit into2π? 2π
π/4 =
8
There are eight pieces like this in the wholepie, whose total area is:
πr2
.
The area of this piece iswhole area
8=
πr2
8
Next slide: What if we cut a different angle? Then how much area do wehave?
Radian measure. 7 / 12
Arc length and sector area
Suppose you have a blueberry pie or radius rand you cut out a piece whose angle is π/4radians.Let’s find out how much area this piece of piehas.
How many of these pieces are in the whole ofthe pie? How many times does π
4 fit into2π? 2π
π/4 = 8
There are eight pieces like this in the wholepie, whose total area is:
πr2
.
The area of this piece iswhole area
8=
πr2
8
Next slide: What if we cut a different angle? Then how much area do wehave?
Radian measure. 7 / 12
Arc length and sector area
Suppose you have a blueberry pie or radius rand you cut out a piece whose angle is π/4radians.Let’s find out how much area this piece of piehas.
How many of these pieces are in the whole ofthe pie? How many times does π
4 fit into2π? 2π
π/4 = 8
There are eight pieces like this in the wholepie, whose total area is:
πr2
.
The area of this piece iswhole area
8=
πr2
8
Next slide: What if we cut a different angle? Then how much area do wehave?
Radian measure. 7 / 12
Arc length and sector area
Suppose you have a blueberry pie or radius rand you cut out a piece whose angle is π/4radians.Let’s find out how much area this piece of piehas.
How many of these pieces are in the whole ofthe pie? How many times does π
4 fit into2π? 2π
π/4 = 8
There are eight pieces like this in the wholepie, whose total area is: πr2.
The area of this piece iswhole area
8=
πr2
8
Next slide: What if we cut a different angle? Then how much area do wehave?
Radian measure. 7 / 12
Arc length and sector area
Suppose you have a blueberry pie or radius rand you cut out a piece whose angle is π/4radians.Let’s find out how much area this piece of piehas.
How many of these pieces are in the whole ofthe pie? How many times does π
4 fit into2π? 2π
π/4 = 8
There are eight pieces like this in the wholepie, whose total area is: πr2.
The area of this piece iswhole area
8=
πr2
8
Next slide: What if we cut a different angle? Then how much area do wehave?
Radian measure. 7 / 12
Arc length and sector area
Suppose you have a blueberry pie or radius rand you cut out a piece whose angle is π/4radians.Let’s find out how much area this piece of piehas.
How many of these pieces are in the whole ofthe pie? How many times does π
4 fit into2π? 2π
π/4 = 8
There are eight pieces like this in the wholepie, whose total area is: πr2.
The area of this piece iswhole area
8= πr2
8
Next slide: What if we cut a different angle? Then how much area do wehave?
Radian measure. 7 / 12
Arc length and sector areaStart with a circle of radius r . Start at the center and draw an angle of θradians. How much area is in this sector?
We know that the area of the whole circle is 2π.
How many times can you fit this sector into the whole circle?
number of radians in a full rotation
θ=
2π
θ
The area of one copy of the sector is
area of circle
number of sectors in the circle=
πr2
2π/θ=
r2θ
2
Theorem
If you start with a circle of radius r and draw a sector of angle θ radiansthen the area of that sector is r2θ
2 .
Radian measure. 8 / 12
Arc length and sector areaStart with a circle of radius r . Start at the center and draw an angle of θradians. How much area is in this sector?
We know that the area of the whole circle is 2π.
How many times can you fit this sector into the whole circle?
number of radians in a full rotation
θ=
2π
θ
The area of one copy of the sector is
area of circle
number of sectors in the circle=
πr2
2π/θ=
r2θ
2
Theorem
If you start with a circle of radius r and draw a sector of angle θ radiansthen the area of that sector is r2θ
2 .
Radian measure. 8 / 12
Arc length and sector areaStart with a circle of radius r . Start at the center and draw an angle of θradians. How much area is in this sector?
We know that the area of the whole circle is 2π.
How many times can you fit this sector into the whole circle?
number of radians in a full rotation
θ=
2π
θ
The area of one copy of the sector is
area of circle
number of sectors in the circle=
πr2
2π/θ=
r2θ
2
Theorem
If you start with a circle of radius r and draw a sector of angle θ radiansthen the area of that sector is r2θ
2 .
Radian measure. 8 / 12
Arc length and sector areaStart with a circle of radius r . Start at the center and draw an angle of θradians. How much area is in this sector?
We know that the area of the whole circle is 2π.
How many times can you fit this sector into the whole circle?
number of radians in a full rotation
θ=
2π
θ
The area of one copy of the sector is
area of circle
number of sectors in the circle=
πr2
2π/θ=
r2θ
2
Theorem
If you start with a circle of radius r and draw a sector of angle θ radiansthen the area of that sector is r2θ
2 .
Radian measure. 8 / 12
Arc length and sector areaStart with a circle of radius r . Start at the center and draw an angle of θradians. How much area is in this sector?
We know that the area of the whole circle is 2π.
How many times can you fit this sector into the whole circle?
number of radians in a full rotation
θ=
2π
θ
The area of one copy of the sector is
area of circle
number of sectors in the circle=
πr2
2π/θ=
r2θ
2
Theorem
If you start with a circle of radius r and draw a sector of angle θ radiansthen the area of that sector is r2θ
2 .
Radian measure. 8 / 12
Arc length and sector areaStart with a circle of radius r . Start at the center and draw an angle of θradians. How much area is in this sector?
We know that the area of the whole circle is 2π.
How many times can you fit this sector into the whole circle?
number of radians in a full rotation
θ=
2π
θ
The area of one copy of the sector is
area of circle
number of sectors in the circle=
πr2
2π/θ=
r2θ
2
Theorem
If you start with a circle of radius r and draw a sector of angle θ radiansthen the area of that sector is r2θ
2 .
Radian measure. 8 / 12
Arc length and sector areaStart with a circle of radius r . Start at the center and draw an angle of θradians. How much area is in this sector?
We know that the area of the whole circle is 2π.
How many times can you fit this sector into the whole circle?
number of radians in a full rotation
θ=
2π
θ
The area of one copy of the sector is
area of circle
number of sectors in the circle=
πr2
2π/θ=
r2θ
2
Theorem
If you start with a circle of radius r and draw a sector of angle θ radiansthen the area of that sector is r2θ
2 .
Radian measure. 8 / 12
Arc length and sector areaStart with a circle of radius r . Start at the center and draw an angle of θradians. How much area is in this sector?
We know that the area of the whole circle is 2π.
How many times can you fit this sector into the whole circle?
number of radians in a full rotation
θ=
2π
θ
The area of one copy of the sector is
area of circle
number of sectors in the circle=
πr2
2π/θ=
r2θ
2
Theorem
If you start with a circle of radius r and draw a sector of angle θ radiansthen the area of that sector is r2θ
2 .
Radian measure. 8 / 12
Arc length and sector areaStart with a circle of radius r . Start at the center and draw an angle of θradians. How much area is in this sector?
We know that the area of the whole circle is 2π.
How many times can you fit this sector into the whole circle?
number of radians in a full rotation
θ=
2π
θ
The area of one copy of the sector is
area of circle
number of sectors in the circle=
πr2
2π/θ=
r2θ
2
Theorem
If you start with a circle of radius r and draw a sector of angle θ radiansthen the area of that sector is r2θ
2 .
Radian measure. 8 / 12
Arc length and sector areaStart with a circle of radius r . Start at the center and draw an angle of θradians. How much area is in this sector?
We know that the area of the whole circle is 2π.
How many times can you fit this sector into the whole circle?
number of radians in a full rotation
θ=
2π
θ
The area of one copy of the sector is
area of circle
number of sectors in the circle=
πr2
2π/θ=
r2θ
2
Theorem
If you start with a circle of radius r and draw a sector of angle θ radiansthen the area of that sector is r2θ
2 .
Radian measure. 8 / 12
sector area and radian measure
Theorem
If you start with a circle of radius r and draw a sector of angle θ then thearea of that sector is r2θ
2 .
Exercises :
1 If you make an angle of 1 radian in a circle or radius 1 then howmuch area is in its sector? r2·θ
2 =
1·12 =
12
2 Draw a right angle in a circle of radius 2. How much area is in itssector? r2·θ
2 =
4·π/42 =
π/2
3 If you make an angle of 1◦ in a circle of radius 20 then how mucharea is in its sector? (You need to convert to radians)r2·θ2 =
400·1· π180
2 =
10π9
Radian measure. 9 / 12
sector area and radian measure
Theorem
If you start with a circle of radius r and draw a sector of angle θ then thearea of that sector is r2θ
2 .
Exercises :
1 If you make an angle of 1 radian in a circle or radius 1 then howmuch area is in its sector? r2·θ
2 =
1·12 =
12
2 Draw a right angle in a circle of radius 2. How much area is in itssector? r2·θ
2 =
4·π/42 =
π/2
3 If you make an angle of 1◦ in a circle of radius 20 then how mucharea is in its sector? (You need to convert to radians)r2·θ2 =
400·1· π180
2 =
10π9
Radian measure. 9 / 12
sector area and radian measure
Theorem
If you start with a circle of radius r and draw a sector of angle θ then thearea of that sector is r2θ
2 .
Exercises :
1 If you make an angle of 1 radian in a circle or radius 1 then howmuch area is in its sector? r2·θ
2 = 1·12 =
12
2 Draw a right angle in a circle of radius 2. How much area is in itssector? r2·θ
2 =
4·π/42 =
π/2
3 If you make an angle of 1◦ in a circle of radius 20 then how mucharea is in its sector? (You need to convert to radians)r2·θ2 =
400·1· π180
2 =
10π9
Radian measure. 9 / 12
sector area and radian measure
Theorem
If you start with a circle of radius r and draw a sector of angle θ then thearea of that sector is r2θ
2 .
Exercises :
1 If you make an angle of 1 radian in a circle or radius 1 then howmuch area is in its sector? r2·θ
2 = 1·12 = 1
2
2 Draw a right angle in a circle of radius 2. How much area is in itssector? r2·θ
2 =
4·π/42 =
π/2
3 If you make an angle of 1◦ in a circle of radius 20 then how mucharea is in its sector? (You need to convert to radians)r2·θ2 =
400·1· π180
2 =
10π9
Radian measure. 9 / 12
sector area and radian measure
Theorem
If you start with a circle of radius r and draw a sector of angle θ then thearea of that sector is r2θ
2 .
Exercises :
1 If you make an angle of 1 radian in a circle or radius 1 then howmuch area is in its sector? r2·θ
2 = 1·12 = 1
2
2 Draw a right angle in a circle of radius 2. How much area is in itssector? r2·θ
2 = 4·π/42 =
π/2
3 If you make an angle of 1◦ in a circle of radius 20 then how mucharea is in its sector? (You need to convert to radians)r2·θ2 =
400·1· π180
2 =
10π9
Radian measure. 9 / 12
sector area and radian measure
Theorem
If you start with a circle of radius r and draw a sector of angle θ then thearea of that sector is r2θ
2 .
Exercises :
1 If you make an angle of 1 radian in a circle or radius 1 then howmuch area is in its sector? r2·θ
2 = 1·12 = 1
2
2 Draw a right angle in a circle of radius 2. How much area is in itssector? r2·θ
2 = 4·π/42 = π/2
3 If you make an angle of 1◦ in a circle of radius 20 then how mucharea is in its sector? (You need to convert to radians)r2·θ2 =
400·1· π180
2 =
10π9
Radian measure. 9 / 12
sector area and radian measure
Theorem
If you start with a circle of radius r and draw a sector of angle θ then thearea of that sector is r2θ
2 .
Exercises :
1 If you make an angle of 1 radian in a circle or radius 1 then howmuch area is in its sector? r2·θ
2 = 1·12 = 1
2
2 Draw a right angle in a circle of radius 2. How much area is in itssector? r2·θ
2 = 4·π/42 = π/2
3 If you make an angle of 1◦ in a circle of radius 20 then how mucharea is in its sector? (You need to convert to radians)r2·θ2 =
400·1· π180
2 =
10π9
Radian measure. 9 / 12
sector area and radian measure
Theorem
If you start with a circle of radius r and draw a sector of angle θ then thearea of that sector is r2θ
2 .
Exercises :
1 If you make an angle of 1 radian in a circle or radius 1 then howmuch area is in its sector? r2·θ
2 = 1·12 = 1
2
2 Draw a right angle in a circle of radius 2. How much area is in itssector? r2·θ
2 = 4·π/42 = π/2
3 If you make an angle of 1◦ in a circle of radius 20 then how mucharea is in its sector? (You need to convert to radians)r2·θ2 =
400·1· π180
2 = 10π9
Radian measure. 9 / 12
Angular speed and linear speed
Definition
Suppose that you have a wheel rotating about its axis.The angular speed of the wheel rotation (in radians per second) is givenby ω = number of radians the wheel rotates through in a second
The linear speed is given byv = how fast a point on the edge of the wheel is moving
How are these related?Suppose a gear of radius 1/3 feet is spinning at 2 radiansper second. Find its linear speed.In one second a point on the circle traces a 2 radian arcof radius 1/3.The length is ` = θ · r = 2 · 1
3 = 2/3.In one second the point has travelled 2/3 feet.
The linear speed is 2/3.
Theorem
The linear and angular speed are related by v = ω · r where r is the radiusand ω is measured in radians per time.
Radian measure. 10 / 12
Angular speed and linear speed
Definition
Suppose that you have a wheel rotating about its axis.The angular speed of the wheel rotation (in radians per second) is givenby ω = number of radians the wheel rotates through in a secondThe linear speed is given byv = how fast a point on the edge of the wheel is moving
How are these related?Suppose a gear of radius 1/3 feet is spinning at 2 radiansper second. Find its linear speed.In one second a point on the circle traces a 2 radian arcof radius 1/3.The length is ` = θ · r = 2 · 1
3 = 2/3.In one second the point has travelled 2/3 feet.
The linear speed is 2/3.
Theorem
The linear and angular speed are related by v = ω · r where r is the radiusand ω is measured in radians per time.
Radian measure. 10 / 12
Angular speed and linear speed
Definition
Suppose that you have a wheel rotating about its axis.The angular speed of the wheel rotation (in radians per second) is givenby ω = number of radians the wheel rotates through in a secondThe linear speed is given byv = how fast a point on the edge of the wheel is moving
How are these related?
Suppose a gear of radius 1/3 feet is spinning at 2 radiansper second. Find its linear speed.In one second a point on the circle traces a 2 radian arcof radius 1/3.The length is ` = θ · r = 2 · 1
3 = 2/3.In one second the point has travelled 2/3 feet.
The linear speed is 2/3.
Theorem
The linear and angular speed are related by v = ω · r where r is the radiusand ω is measured in radians per time.
Radian measure. 10 / 12
Angular speed and linear speed
Definition
Suppose that you have a wheel rotating about its axis.The angular speed of the wheel rotation (in radians per second) is givenby ω = number of radians the wheel rotates through in a secondThe linear speed is given byv = how fast a point on the edge of the wheel is moving
How are these related?Suppose a gear of radius 1/3 feet is spinning at 2 radiansper second. Find its linear speed.
In one second a point on the circle traces a 2 radian arcof radius 1/3.The length is ` = θ · r = 2 · 1
3 = 2/3.In one second the point has travelled 2/3 feet.
The linear speed is 2/3.
Theorem
The linear and angular speed are related by v = ω · r where r is the radiusand ω is measured in radians per time.
Radian measure. 10 / 12
Angular speed and linear speed
Definition
Suppose that you have a wheel rotating about its axis.The angular speed of the wheel rotation (in radians per second) is givenby ω = number of radians the wheel rotates through in a secondThe linear speed is given byv = how fast a point on the edge of the wheel is moving
How are these related?Suppose a gear of radius 1/3 feet is spinning at 2 radiansper second. Find its linear speed.In one second a point on the circle traces a 2 radian arcof radius 1/3.
The length is ` = θ · r = 2 · 13 = 2/3.
In one second the point has travelled 2/3 feet.
The linear speed is 2/3.
Theorem
The linear and angular speed are related by v = ω · r where r is the radiusand ω is measured in radians per time.
Radian measure. 10 / 12
Angular speed and linear speed
Definition
Suppose that you have a wheel rotating about its axis.The angular speed of the wheel rotation (in radians per second) is givenby ω = number of radians the wheel rotates through in a secondThe linear speed is given byv = how fast a point on the edge of the wheel is moving
How are these related?Suppose a gear of radius 1/3 feet is spinning at 2 radiansper second. Find its linear speed.In one second a point on the circle traces a 2 radian arcof radius 1/3.The length is ` = θ · r = 2 · 1
3 = 2/3.
In one second the point has travelled 2/3 feet.
The linear speed is 2/3.
Theorem
The linear and angular speed are related by v = ω · r where r is the radiusand ω is measured in radians per time.
Radian measure. 10 / 12
Angular speed and linear speed
Definition
Suppose that you have a wheel rotating about its axis.The angular speed of the wheel rotation (in radians per second) is givenby ω = number of radians the wheel rotates through in a secondThe linear speed is given byv = how fast a point on the edge of the wheel is moving
How are these related?Suppose a gear of radius 1/3 feet is spinning at 2 radiansper second. Find its linear speed.In one second a point on the circle traces a 2 radian arcof radius 1/3.The length is ` = θ · r = 2 · 1
3 = 2/3.In one second the point has travelled 2/3 feet.The linear speed is 2/3.
Theorem
The linear and angular speed are related by v = ω · r where r is the radiusand ω is measured in radians per time.
Radian measure. 10 / 12
Angular speed and linear speed
Theorem
The linear and angular speed are related by v = ω · r where r is the radiusand ω is measured in radians per second or radians per minute or radiansper hour.
If a gear of radius 5 inches is spinning at 6 revolutions per minute. Whatis its linear speed?
Step 1: Compute angular velocity in radians per minute. One revolution is2π radians: The angular velocity is
ω = 6 revolutions per minute= 6 · 2π radians per minute= 12π radians per minute
Step 2: Get linear velocity:
v = (radius) · ω = 5 · 12 feet/minute = 60 feet/minute = 1 foot/second
Radian measure. 11 / 12
Angular speed and linear speed
Theorem
The linear and angular speed are related by v = ω · r where r is the radiusand ω is measured in radians per second or radians per minute or radiansper hour.
If a gear of radius 5 inches is spinning at 6 revolutions per minute. Whatis its linear speed?Step 1: Compute angular velocity in radians per minute. One revolution is2π radians: The angular velocity is
ω = 6 revolutions per minute=
6 · 2π radians per minute= 12π radians per minute
Step 2: Get linear velocity:
v = (radius) · ω = 5 · 12 feet/minute = 60 feet/minute = 1 foot/second
Radian measure. 11 / 12
Angular speed and linear speed
Theorem
The linear and angular speed are related by v = ω · r where r is the radiusand ω is measured in radians per second or radians per minute or radiansper hour.
If a gear of radius 5 inches is spinning at 6 revolutions per minute. Whatis its linear speed?Step 1: Compute angular velocity in radians per minute. One revolution is2π radians: The angular velocity is
ω = 6 revolutions per minute= 6 · 2π radians per minute=
12π radians per minute
Step 2: Get linear velocity:
v = (radius) · ω = 5 · 12 feet/minute = 60 feet/minute = 1 foot/second
Radian measure. 11 / 12
Angular speed and linear speed
Theorem
The linear and angular speed are related by v = ω · r where r is the radiusand ω is measured in radians per second or radians per minute or radiansper hour.
If a gear of radius 5 inches is spinning at 6 revolutions per minute. Whatis its linear speed?Step 1: Compute angular velocity in radians per minute. One revolution is2π radians: The angular velocity is
ω = 6 revolutions per minute= 6 · 2π radians per minute= 12π radians per minute
Step 2: Get linear velocity:
v = (radius) · ω = 5 · 12 feet/minute = 60 feet/minute = 1 foot/second
Radian measure. 11 / 12
Angular speed and linear speed
Theorem
The linear and angular speed are related by v = ω · r where r is the radiusand ω is measured in radians per second or radians per minute or radiansper hour.
If a gear of radius 5 inches is spinning at 6 revolutions per minute. Whatis its linear speed?Step 1: Compute angular velocity in radians per minute. One revolution is2π radians: The angular velocity is
ω = 6 revolutions per minute= 6 · 2π radians per minute= 12π radians per minute
Step 2: Get linear velocity:
v = (radius) · ω =
5 · 12 feet/minute = 60 feet/minute = 1 foot/second
Radian measure. 11 / 12
Angular speed and linear speed
Theorem
The linear and angular speed are related by v = ω · r where r is the radiusand ω is measured in radians per second or radians per minute or radiansper hour.
If a gear of radius 5 inches is spinning at 6 revolutions per minute. Whatis its linear speed?Step 1: Compute angular velocity in radians per minute. One revolution is2π radians: The angular velocity is
ω = 6 revolutions per minute= 6 · 2π radians per minute= 12π radians per minute
Step 2: Get linear velocity:
v = (radius) · ω = 5 · 12 feet/minute =
60 feet/minute = 1 foot/second
Radian measure. 11 / 12
Angular speed and linear speed
Theorem
The linear and angular speed are related by v = ω · r where r is the radiusand ω is measured in radians per second or radians per minute or radiansper hour.
If a gear of radius 5 inches is spinning at 6 revolutions per minute. Whatis its linear speed?Step 1: Compute angular velocity in radians per minute. One revolution is2π radians: The angular velocity is
ω = 6 revolutions per minute= 6 · 2π radians per minute= 12π radians per minute
Step 2: Get linear velocity:
v = (radius) · ω = 5 · 12 feet/minute = 60 feet/minute =
1 foot/second
Radian measure. 11 / 12
Angular speed and linear speed
Theorem
The linear and angular speed are related by v = ω · r where r is the radiusand ω is measured in radians per second or radians per minute or radiansper hour.
If a gear of radius 5 inches is spinning at 6 revolutions per minute. Whatis its linear speed?Step 1: Compute angular velocity in radians per minute. One revolution is2π radians: The angular velocity is
ω = 6 revolutions per minute= 6 · 2π radians per minute= 12π radians per minute
Step 2: Get linear velocity:
v = (radius) · ω = 5 · 12 feet/minute = 60 feet/minute = 1 foot/second
Radian measure. 11 / 12
exit quiz:
A gear of radius 2 is spinning 30 degrees per minute.
1 What is its angular speed in radians per minute?
2 What is its linear velocity?
3 This gear is connected to another gear of radius 1 so that is has thesame linear velocity. What is its angular velocity, in radians perminute and in degrees per minute.
Radian measure. 12 / 12
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