Radian measure.

() Radian measure. 1 / 11

Measuring angles

When someone talks about an angle of 72◦ how do you know what theymean?

You start by remembering that 90◦ corresponds to a right angle.

Since 72 = 90 · 45 you scale back this angle by a factor of 4

5 .

This is 72◦.Does this seem arbitrary to you?Why is 90◦ a right angle?

() Radian measure. 2 / 11

Measuring angles

When someone talks about an angle of 72◦ how do you know what theymean?You start by remembering that 90◦ corresponds to a right angle.

Since 72 = 90 · 45 you scale back this angle by a factor of 4

5 .

This is 72◦.Does this seem arbitrary to you?Why is 90◦ a right angle?

() Radian measure. 2 / 11

Measuring angles

When someone talks about an angle of 72◦ how do you know what theymean?You start by remembering that 90◦ corresponds to a right angle.

Since 72 = 90 · 45 you scale back this angle by a factor of 4

5 .

This is 72◦.Does this seem arbitrary to you?Why is 90◦ a right angle?

() Radian measure. 2 / 11

Measuring angles

When someone talks about an angle of 72◦ how do you know what theymean?You start by remembering that 90◦ corresponds to a right angle.

Since 72 = 90 · 45 you scale back this angle by a factor of 4

5 .

This is 72◦.

Does this seem arbitrary to you?Why is 90◦ a right angle?

() Radian measure. 2 / 11

Measuring angles

When someone talks about an angle of 72◦ how do you know what theymean?You start by remembering that 90◦ corresponds to a right angle.

Since 72 = 90 · 45 you scale back this angle by a factor of 4

5 .

This is 72◦.Does this seem arbitrary to you?Why is 90◦ a right angle?

() Radian measure. 2 / 11

Radian measureImagine 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 angles are these?

2 cm

2 cm

1.5 cm

6 cm

() Radian measure. 3 / 11

Radian measureImagine 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 angles are these?

2 cm

2 cm

1.5 cm

6 cm

() Radian measure. 3 / 11

Radian measureImagine 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 angles are these?

2 cm

2 cm

1.5 cm

6 cm

() Radian measure. 3 / 11

Radian measureImagine 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 angles are these?

2 cm

2 cm

1.5 cm

6 cm

() Radian measure. 3 / 11

Radian measureImagine 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 angles are these?

2 cm

2 cm

1.5 cm

6 cm

() Radian measure. 3 / 11

Radian measureImagine 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 angles are these?

2 cm

2 cm

1.5 cm

6 cm

() Radian measure. 3 / 11

Radian measureImagine 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 angles are these?

2 cm

2 cm

1.5 cm

6 cm

() Radian measure. 3 / 11

Converting between radians and degrees

Definition

An arc of length s on a circle of radius r covers an angle of sr radians.

What angle corresponds to walking all the way around a circle?In degrees?

360◦ is one full revolutionIn radians?Imagine starting on a circle of radius 1. How long an arc do you have todraw to get all the way around? What is the circumference of a circle ofradius 1?The circumference of a circle of radius 1 is 2π. This angle is

arc length

radius=

2π

1= 2π radians

360◦ is equal to 2π radians.

() Radian measure. 4 / 11

Converting between radians and degrees

Definition

An arc of length s on a circle of radius r covers an angle of sr radians.

What angle corresponds to walking all the way around a circle?In degrees? 360◦ is one full revolution

In radians?Imagine starting on a circle of radius 1. How long an arc do you have todraw to get all the way around? What is the circumference of a circle ofradius 1?The circumference of a circle of radius 1 is 2π. This angle is

arc length

radius=

2π

1= 2π radians

360◦ is equal to 2π radians.

() Radian measure. 4 / 11

Converting between radians and degrees

Definition

An arc of length s on a circle of radius r covers an angle of sr radians.

What angle corresponds to walking all the way around a circle?In degrees? 360◦ is one full revolutionIn radians?

Imagine starting on a circle of radius 1. How long an arc do you have todraw to get all the way around? What is the circumference of a circle ofradius 1?The circumference of a circle of radius 1 is 2π. This angle is

arc length

radius=

2π

1= 2π radians

360◦ is equal to 2π radians.

() Radian measure. 4 / 11

Converting between radians and degrees

Definition

An arc of length s on a circle of radius r covers an angle of sr radians.

What angle corresponds to walking all the way around a circle?In degrees? 360◦ is one full revolutionIn radians?Imagine starting on a circle of radius 1. How long an arc do you have todraw to get all the way around? What is the circumference of a circle ofradius 1?

The circumference of a circle of radius 1 is 2π. This angle is

arc length

radius=

2π

1= 2π radians

360◦ is equal to 2π radians.

() Radian measure. 4 / 11

Converting between radians and degrees

Definition

An arc of length s on a circle of radius r covers an angle of sr radians.

What angle corresponds to walking all the way around a circle?In degrees? 360◦ is one full revolutionIn radians?Imagine starting on a circle of radius 1. How long an arc do you have todraw to get all the way around? What is the circumference of a circle ofradius 1?The circumference of a circle of radius 1 is 2π. This angle is

arc length

radius=

2π

1= 2π radians

360◦ is equal to 2π radians.

() Radian measure. 4 / 11

Converting between radians and degrees

Definition

An arc of length s on a circle of radius r covers an angle of sr radians.

What angle corresponds to walking all the way around a circle?In degrees? 360◦ is one full revolutionIn radians?Imagine starting on a circle of radius 1. How long an arc do you have todraw to get all the way around? What is the circumference of a circle ofradius 1?The circumference of a circle of radius 1 is 2π. This angle is

arc length

radius=

2π

1= 2π radians

360◦ is equal to 2π radians.

() Radian measure. 4 / 11

Converting between radians and degrees

360◦ is equal to 2π radians.

0◦ is

0

radians. (Fill in the blank)

The relationship between degrees and radians is linear. (If you doublethe radian measure then you double the angle measure)

What linear function sends 0 7→ 0 and 360 7→ 2π?Slope= 2π

360 = π180 . Passes through (0, 0).

Theorem

An angle of d degrees is equal to an angle of d · π180 radians.

An angle of r radians is deal to an angle of r · 180π degrees.

Group quiz: (Will be collected)

1 A right angle is 90◦. What is its radian measure?

2 What radian measure corresponds to doing a 180 degree rotation?

3 If a radical skateboarder completes a 720◦ rotation in the air. What isthe radian measure of his rotation?

() Radian measure. 5 / 11

Converting between radians and degrees

360◦ is equal to 2π radians.

0◦ is

0

radians. (Fill in the blank)

The relationship between degrees and radians is linear. (If you doublethe radian measure then you double the angle measure)

What linear function sends 0 7→ 0 and 360 7→ 2π?Slope= 2π

360 = π180 . Passes through (0, 0).

Theorem

An angle of d degrees is equal to an angle of d · π180 radians.

An angle of r radians is deal to an angle of r · 180π degrees.

Group quiz: (Will be collected)

1 A right angle is 90◦. What is its radian measure?

2 What radian measure corresponds to doing a 180 degree rotation?

3 If a radical skateboarder completes a 720◦ rotation in the air. What isthe radian measure of his rotation?

() Radian measure. 5 / 11

Converting between radians and degrees

360◦ is equal to 2π radians.

0◦ is 0 radians. (Fill in the blank)

The relationship between degrees and radians is linear. (If you doublethe radian measure then you double the angle measure)

What linear function sends 0 7→ 0 and 360 7→ 2π?Slope= 2π

360 = π180 . Passes through (0, 0).

Theorem

An angle of d degrees is equal to an angle of d · π180 radians.

An angle of r radians is deal to an angle of r · 180π degrees.

Group quiz: (Will be collected)

1 A right angle is 90◦. What is its radian measure?

2 What radian measure corresponds to doing a 180 degree rotation?

3 If a radical skateboarder completes a 720◦ rotation in the air. What isthe radian measure of his rotation?

() Radian measure. 5 / 11

Converting between radians and degrees

360◦ is equal to 2π radians.

0◦ is 0 radians. (Fill in the blank)

The relationship between degrees and radians is linear. (If you doublethe radian measure then you double the angle measure)

What linear function sends 0 7→ 0 and 360 7→ 2π?Slope= 2π

360 = π180 . Passes through (0, 0).

Theorem

An angle of d degrees is equal to an angle of d · π180 radians.

An angle of r radians is deal to an angle of r · 180π degrees.

Group quiz: (Will be collected)

1 A right angle is 90◦. What is its radian measure?

2 What radian measure corresponds to doing a 180 degree rotation?

3 If a radical skateboarder completes a 720◦ rotation in the air. What isthe radian measure of his rotation?

() Radian measure. 5 / 11

Converting between radians and degrees

360◦ is equal to 2π radians.

0◦ is 0 radians. (Fill in the blank)

The relationship between degrees and radians is linear. (If you doublethe radian measure then you double the angle measure)

What linear function sends 0 7→ 0 and 360 7→ 2π?

Slope= 2π360 = π

180 . Passes through (0, 0).

Theorem

An angle of d degrees is equal to an angle of d · π180 radians.

An angle of r radians is deal to an angle of r · 180π degrees.

Group quiz: (Will be collected)

1 A right angle is 90◦. What is its radian measure?

2 What radian measure corresponds to doing a 180 degree rotation?

3 If a radical skateboarder completes a 720◦ rotation in the air. What isthe radian measure of his rotation?

() Radian measure. 5 / 11

Converting between radians and degrees

360◦ is equal to 2π radians.

0◦ is 0 radians. (Fill in the blank)

The relationship between degrees and radians is linear. (If you doublethe radian measure then you double the angle measure)

What linear function sends 0 7→ 0 and 360 7→ 2π?Slope= 2π

360 = π180 . Passes through (0, 0).

Theorem

An angle of d degrees is equal to an angle of d · π180 radians.

An angle of r radians is deal to an angle of r · 180π degrees.

Group quiz: (Will be collected)

1 A right angle is 90◦. What is its radian measure?

2 What radian measure corresponds to doing a 180 degree rotation?

3 If a radical skateboarder completes a 720◦ rotation in the air. What isthe radian measure of his rotation?

() Radian measure. 5 / 11

Converting between radians and degrees

360◦ is equal to 2π radians.

0◦ is 0 radians. (Fill in the blank)

The relationship between degrees and radians is linear. (If you doublethe radian measure then you double the angle measure)

What linear function sends 0 7→ 0 and 360 7→ 2π?Slope= 2π

360 = π180 . Passes through (0, 0).

Theorem

An angle of d degrees is equal to an angle of d · π180 radians.

An angle of r radians is deal to an angle of r · 180π degrees.

Group quiz: (Will be collected)

1 A right angle is 90◦. What is its radian measure?

2 What radian measure corresponds to doing a 180 degree rotation?

3 If a radical skateboarder completes a 720◦ rotation in the air. What isthe radian measure of his rotation?

() Radian measure. 5 / 11

Arc length and sector area

Suppose you have a blueberry pie or radius rand you cut out a piece whose angle is π/4radians. How much area has this piece of pie?

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 the the pie,whose area is:

πr2

.

The are of this piece iswhole area

8= πr2

8

What if we cut a different angle?

() Radian measure. 6 / 11

Arc length and sector area

Suppose you have a blueberry pie or radius rand you cut out a piece whose angle is π/4radians. How much area has this piece of pie?

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 the the pie,whose area is:

πr2

.

The are of this piece iswhole area

8= πr2

8

What if we cut a different angle?

() Radian measure. 6 / 11

Arc length and sector area

Suppose you have a blueberry pie or radius rand you cut out a piece whose angle is π/4radians. How much area has this piece of pie?

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 the the pie,whose area is:

πr2

.

The are of this piece iswhole area

8= πr2

8

What if we cut a different angle?

() Radian measure. 6 / 11

Arc length and sector area

Suppose you have a blueberry pie or radius rand you cut out a piece whose angle is π/4radians. How much area has this piece of pie?

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 the the pie,whose area is:

πr2

.

The are of this piece iswhole area

8= πr2

8

What if we cut a different angle?

() Radian measure. 6 / 11

Arc length and sector area

Suppose you have a blueberry pie or radius rand you cut out a piece whose angle is π/4radians. How much area has this piece of pie?

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 the the pie,whose area is:

πr2

.

The are of this piece iswhole area

8= πr2

8

What if we cut a different angle?

() Radian measure. 6 / 11

Arc length and sector area

Suppose you have a blueberry pie or radius rand you cut out a piece whose angle is π/4radians. How much area has this piece of pie?

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 the the pie,whose area is:

πr2

.

The are of this piece iswhole area

8= πr2

8

What if we cut a different angle?

() Radian measure. 6 / 11

Arc length and sector area

Suppose you have a blueberry pie or radius rand you cut out a piece whose angle is π/4radians. How much area has this piece of pie?

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 the the pie,whose area is: πr2.

The are of this piece iswhole area

8= πr2

8

What if we cut a different angle?

() Radian measure. 6 / 11

Arc length and sector area

Suppose you have a blueberry pie or radius rand you cut out a piece whose angle is π/4radians. How much area has this piece of pie?

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 the the pie,whose area is: πr2.

The are of this piece iswhole area

8= πr2

8

What if we cut a different angle?

() Radian measure. 6 / 11

Arc length and sector area

Suppose you have a blueberry pie or radius rand you cut out a piece whose angle is π/4radians. How much area has this piece of pie?

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 the the pie,whose area is: πr2.

The are of this piece iswhole area

8= πr2

8

What if we cut a different angle?

() Radian measure. 6 / 11

Arc length and sector areaStart with a circle of radius r . Start at the center and draw an angle of θ.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 then thearea of that sector is r2θ

2 .

() Radian measure. 7 / 11

Arc length and sector areaStart with a circle of radius r . Start at the center and draw an angle of θ.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 then thearea of that sector is r2θ

2 .

() Radian measure. 7 / 11

Arc length and sector areaStart with a circle of radius r . Start at the center and draw an angle of θ.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 then thearea of that sector is r2θ

2 .

() Radian measure. 7 / 11

Arc length and sector areaStart with a circle of radius r . Start at the center and draw an angle of θ.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 then thearea of that sector is r2θ

2 .

() Radian measure. 7 / 11

Arc length and sector areaStart with a circle of radius r . Start at the center and draw an angle of θ.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 then thearea of that sector is r2θ

2 .

() Radian measure. 7 / 11

Arc length and sector areaStart with a circle of radius r . Start at the center and draw an angle of θ.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 then thearea of that sector is r2θ

2 .

() Radian measure. 7 / 11

Arc length and sector areaStart with a circle of radius r . Start at the center and draw an angle of θ.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 then thearea of that sector is r2θ

2 .

() Radian measure. 7 / 11

Arc length and sector areaStart with a circle of radius r . Start at the center and draw an angle of θ.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 then thearea of that sector is r2θ

2 .

() Radian measure. 7 / 11

Arc length and sector areaStart with a circle of radius r . Start at the center and draw an angle of θ.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 then thearea of that sector is r2θ

2 .

() Radian measure. 7 / 11

sector area and radian masure

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 .

Group Work:

1 Draw a right angle in a circle of radius 2. How much area is in itssector?

r2·θ2 =

4·π/42 =

π/2

2 If you make an angle of 1◦ in a circle of radius 20 then how much

area is in its sector?

r2·θ2 =

400·1· π180

2 =

10π9

3 If you make an angle of 1 radian in a circle or radius 1 then howmuch area is in its sector?

r2·θ2 =

1·12 =

12

() Radian measure. 8 / 11

sector area and radian masure

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 .

Group Work:

1 Draw a right angle in a circle of radius 2. How much area is in itssector? r2·θ

2 =

4·π/42 =

π/2

2 If you make an angle of 1◦ in a circle of radius 20 then how much

area is in its sector?

r2·θ2 =

400·1· π180

2 =

10π9

3 If you make an angle of 1 radian in a circle or radius 1 then howmuch area is in its sector?

r2·θ2 =

1·12 =

12

() Radian measure. 8 / 11

sector area and radian masure

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 .

Group Work:

1 Draw a right angle in a circle of radius 2. How much area is in itssector? r2·θ

2 = 4·π/42 =

π/2

2 If you make an angle of 1◦ in a circle of radius 20 then how much

area is in its sector?

r2·θ2 =

400·1· π180

2 =

10π9

3 If you make an angle of 1 radian in a circle or radius 1 then howmuch area is in its sector?

r2·θ2 =

1·12 =

12

() Radian measure. 8 / 11

sector area and radian masure

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 .

Group Work:

1 Draw a right angle in a circle of radius 2. How much area is in itssector? r2·θ

2 = 4·π/42 = π/2

2 If you make an angle of 1◦ in a circle of radius 20 then how much

area is in its sector?

r2·θ2 =

400·1· π180

2 =

10π9

3 If you make an angle of 1 radian in a circle or radius 1 then howmuch area is in its sector?

r2·θ2 =

1·12 =

12

() Radian measure. 8 / 11

sector area and radian masure

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 .

Group Work:

1 Draw a right angle in a circle of radius 2. How much area is in itssector? r2·θ

2 = 4·π/42 = π/2

2 If you make an angle of 1◦ in a circle of radius 20 then how much

area is in its sector? r2·θ2 =

400·1· π180

2 =

10π9

3 If you make an angle of 1 radian in a circle or radius 1 then howmuch area is in its sector?

r2·θ2 =

1·12 =

12

() Radian measure. 8 / 11

sector area and radian masure

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 .

Group Work:

1 Draw a right angle in a circle of radius 2. How much area is in itssector? r2·θ

2 = 4·π/42 = π/2

2 If you make an angle of 1◦ in a circle of radius 20 then how much

area is in its sector? r2·θ2 =

400·1· π180

2 =

10π9

3 If you make an angle of 1 radian in a circle or radius 1 then howmuch area is in its sector?

r2·θ2 =

1·12 =

12

() Radian measure. 8 / 11

sector area and radian masure

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 .

Group Work:

1 Draw a right angle in a circle of radius 2. How much area is in itssector? r2·θ

2 = 4·π/42 = π/2

2 If you make an angle of 1◦ in a circle of radius 20 then how much

area is in its sector? r2·θ2 =

400·1· π180

2 = 10π9

3 If you make an angle of 1 radian in a circle or radius 1 then howmuch area is in its sector?

r2·θ2 =

1·12 =

12

() Radian measure. 8 / 11

sector area and radian masure

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 .

Group Work:

1 Draw a right angle in a circle of radius 2. How much area is in itssector? r2·θ

2 = 4·π/42 = π/2

2 If you make an angle of 1◦ in a circle of radius 20 then how much

area is in its sector? r2·θ2 =

400·1· π180

2 = 10π9

3 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

() Radian measure. 8 / 11

sector area and radian masure

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 .

Group Work:

1 Draw a right angle in a circle of radius 2. How much area is in itssector? r2·θ

2 = 4·π/42 = π/2

2 If you make an angle of 1◦ in a circle of radius 20 then how much

area is in its sector? r2·θ2 =

400·1· π180

2 = 10π9

3 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

() Radian measure. 8 / 11

sector area and radian masure

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 .

Group Work:

1 Draw a right angle in a circle of radius 2. How much area is in itssector? r2·θ

2 = 4·π/42 = π/2

2 If you make an angle of 1◦ in a circle of radius 20 then how much

area is in its sector? r2·θ2 =

400·1· π180

2 = 10π9

3 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

() Radian measure. 8 / 11

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 quickly the arc drawn by a point on the circle grows.

How are these related?Suppose a gear of radius 1/3 is spinning at 2 radiansper second. Find its linear speed.In one second a point on the circle traces a 2 radianarc of radius 1/3.The length is ` = θ · r = 2 · 1

3 = 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. 9 / 11

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 quickly the arc drawn by a point on the circle grows.

How are these related?Suppose a gear of radius 1/3 is spinning at 2 radiansper second. Find its linear speed.In one second a point on the circle traces a 2 radianarc of radius 1/3.The length is ` = θ · r = 2 · 1

3 = 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. 9 / 11

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 quickly the arc drawn by a point on the circle grows.

How are these related?

Suppose a gear of radius 1/3 is spinning at 2 radiansper second. Find its linear speed.In one second a point on the circle traces a 2 radianarc of radius 1/3.The length is ` = θ · r = 2 · 1

3 = 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. 9 / 11

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 quickly the arc drawn by a point on the circle grows.

How are these related?Suppose a gear of radius 1/3 is spinning at 2 radiansper second. Find its linear speed.

In one second a point on the circle traces a 2 radianarc of radius 1/3.The length is ` = θ · r = 2 · 1

3 = 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. 9 / 11

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 quickly the arc drawn by a point on the circle grows.

How are these related?Suppose a gear of radius 1/3 is spinning at 2 radiansper second. Find its linear speed.In one second a point on the circle traces a 2 radianarc of radius 1/3.

The length is ` = θ · r = 2 · 13 = 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. 9 / 11

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 quickly the arc drawn by a point on the circle grows.

How are these related?Suppose a gear of radius 1/3 is spinning at 2 radiansper second. Find its linear speed.In one second a point on the circle traces a 2 radianarc of radius 1/3.The length is ` = θ · r = 2 · 1

3 = 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. 9 / 11

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 quickly the arc drawn by a point on the circle grows.

How are these related?Suppose a gear of radius 1/3 is spinning at 2 radiansper second. Find its linear speed.In one second a point on the circle traces a 2 radianarc of radius 1/3.The length is ` = θ · r = 2 · 1

3 = 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. 9 / 11

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 time.

If a gear of radius 5 inches is spinning at 6 revolutions per minute. Whatis its linear speed?

Step 1: Compute angular velocity in terms of radians One revolution is 2πradians: The angular velocity is

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. 10 / 11

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 time.

If a gear of radius 5 inches is spinning at 6 revolutions per minute. Whatis its linear speed?Step 1: Compute angular velocity in terms of radians One revolution is 2πradians: The angular velocity is

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. 10 / 11

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 time.

If a gear of radius 5 inches is spinning at 6 revolutions per minute. Whatis its linear speed?Step 1: Compute angular velocity in terms of radians One revolution is 2πradians: The angular velocity is

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. 10 / 11

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 time.

If a gear of radius 5 inches is spinning at 6 revolutions per minute. Whatis its linear speed?Step 1: Compute angular velocity in terms of radians One revolution is 2πradians: The angular velocity is

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. 10 / 11

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 time.

If a gear of radius 5 inches is spinning at 6 revolutions per minute. Whatis its linear speed?Step 1: Compute angular velocity in terms of radians One revolution is 2πradians: The angular velocity is

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. 10 / 11

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 time.

If a gear of radius 5 inches is spinning at 6 revolutions per minute. Whatis its linear speed?Step 1: Compute angular velocity in terms of radians One revolution is 2πradians: The angular velocity is

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. 10 / 11

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 time.

If a gear of radius 5 inches is spinning at 6 revolutions per minute. Whatis its linear speed?Step 1: Compute angular velocity in terms of radians One revolution is 2πradians: The angular velocity is

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. 10 / 11

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. 11 / 11