Radian–Measurement Applications

r = 1

The unit circle is the circle

centered at (0, 0) with radius 1.

It’s the graph of x2 + y2 = 1.

The radian measurement of the

angle is the length of the arc

that the angle cuts off

from the unit circle.

(1, 0)

Arc length as angle

measurement for

Radian–Measurement Applications

r = 1

The unit circle is the circle

centered at (0, 0) with radius 1.

It’s the graph of x2 + y2 = 1.

The radian measurement of the

angle is the length of the arc

that the angle cuts off

from the unit circle.

(1, 0)

Arc length as angle

measurement for

Radian–Measurement Applications

r = 1

Conversions between Degree and Radian

The unit circle is the circle

centered at (0, 0) with radius 1.

It’s the graph of x2 + y2 = 1.

The radian measurement of the

angle is the length of the arc

that the angle cuts off

from the unit circle.

180o = π rad

(1, 0)

Arc length as angle

measurement for

Radian–Measurement Applications

r = 1

Conversions between Degree and Radian

The unit circle is the circle

centered at (0, 0) with radius 1.

It’s the graph of x2 + y2 = 1.

The radian measurement of the

angle is the length of the arc

that the angle cuts off

from the unit circle.

180o = π rad 90o = radπ

2

(1, 0)

Arc length as angle

measurement for

Radian–Measurement Applications

r = 1

Conversions between Degree and Radian

The unit circle is the circle

centered at (0, 0) with radius 1.

It’s the graph of x2 + y2 = 1.

The radian measurement of the

angle is the length of the arc

that the angle cuts off

from the unit circle.

180o = π rad 90o = radπ

2 60o = radπ

3 45o = radπ

4

(1, 0)

Arc length as angle

measurement for

Radian–Measurement Applications

r = 1

Conversions between Degree and Radian

π

180 π

180o

The unit circle is the circle

centered at (0, 0) with radius 1.

It’s the graph of x2 + y2 = 1.

The radian measurement of the

angle is the length of the arc

that the angle cuts off

from the unit circle.

1o = 0.0175 rad 1 rad = 57o ‘

180o = π rad 90o = radπ

2 60o = radπ

3 45o = radπ

4

(1, 0)

Arc length as angle

measurement for

Radian–Measurement Applications

r = 1

Conversions between Degree and Radian

π

180 π

180o

The unit circle is the circle

centered at (0, 0) with radius 1.

It’s the graph of x2 + y2 = 1.

The radian measurement of the

angle is the length of the arc

that the angle cuts off

from the unit circle.

1o = 0.0175 rad 1 rad = 57o ‘

180o = π rad 90o = radπ

2 60o = radπ

3 45o = radπ

4

(1, 0)

The advantage of using circular–lengths (radians)

to measure angles is that formulas concerning circles

may be stated with greater simplicity.

An angle based at the center of

a circle is called a central angle.

r

Arc Length Formula

Radian Arc-Length Formula

Given a circle of radius = r

and a central angle in radian,

let L = length of the arc cuts off by

the angle ,

An angle based at the center of

a circle is called a central angle.

r

Arc Length Formula

Radian Arc-Length Formula

Given a circle of radius = r

and a central angle in radian,

let L = length of the arc cuts off by

the angle , then

L = r

An angle based at the center of

a circle is called a central angle.

r

Arc Length Formula

L = r

L= arc length

Radian Arc-Length Formula

Given a circle of radius = r

and a central angle in radian,

let L = length of the arc cuts off by

the angle , then

L = r

An angle based at the center of

a circle is called a central angle.

r

Arc Length Formula

L = r

L= arc length

This formula is based on the following proportion

which demonstrates the advantage of using radian.

Radian Arc-Length Formula

Given a circle of radius = r

and a central angle in radian,

let L = length of the arc cuts off by

the angle , then

L = r

An angle based at the center of

a circle is called a central angle.

r

Arc Length Formula

L = r

L= arc length

This formula is based on the following proportion

which demonstrates the advantage of using radian.

arc length : circumference = L : 2πr = : 2π (in rad)

2πr

L =

2π

or

Radian Arc-Length Formula

Given a circle of radius = r

and a central angle in radian,

let L = length of the arc cuts off by

the angle , then

L = r

An angle based at the center of

a circle is called a central angle.

r

Arc Length Formula

L = r

L= arc length

This formula is based on the following proportion

which demonstrates the advantage of using radian.

arc length : circumference = L : 2πr = : 2π (in rad)

2πr

L =

2π

or clear denominators, we´ve

L = r

Arc Length Formula

Example A.

a. A slice of pizza with central angle of 50o is cut from

a 36”-diamter pizza, what is the length of its crust?

Arc Length Formula

Example A.

a. A slice of pizza with central angle of 50o is cut from

a 36”-diamter pizza, what is the length of its crust?

50o

18

?

a.

Arc Length Formula

Example A.

a. A slice of pizza with central angle of 50o is cut from

a 36”-diamter pizza, what is the length of its crust?

The point here is that we need radian!

50o

18

?

a.

Arc Length Formula

Example A.

a. A slice of pizza with central angle of 50o is cut from

a 36”-diamter pizza, what is the length of its crust?

The point here is that we need radian!

50o = 50*(π/180 rad) = 5π/18 rad,

50o

18

?

a.

Arc Length Formula

Example A.

a. A slice of pizza with central angle of 50o is cut from

a 36”-diamter pizza, what is the length of its crust?

The point here is that we need radian!

50o = 50*(π/180 rad) = 5π/18 rad, with r = 18,

the length of the crust is r = 18·

5π

18 = 5π 15.7″

50o

18

?

a.

Arc Length Formula

b. A slice of pizza cut from a pizza with 32” diameter

has a crust measured 12”. Find its central angle.

Give the answer in radian and degree.

Example A.

a. A slice of pizza with central angle of 50o is cut from

a 36”-diamter pizza, what is the length of its crust?

The point here is that we need radian!

50o = 50*(π/180 rad) = 5π/18 rad, with r = 18,

the length of the crust is r = 18·

5π

18 = 5π 15.7″

50o

18

?

16

?

12

a. b.

Arc Length Formula

Given that r = 16″ and L = 12″,

therefore 16· = 12

b. A slice of pizza cut from a pizza with 32” diameter

has a crust measured 12”. Find its central angle.

Give the answer in radian and degree.

Example A.

a. A slice of pizza with central angle of 50o is cut from

a 36”-diamter pizza, what is the length of its crust?

The point here is that we need radian!

50o = 50*(π/180 rad) = 5π/18 rad, with r = 18,

the length of the crust is r = 18·

5π

18 = 5π 15.7″

50o

18

?

16

?

12

a. b.

Arc Length Formula

Given that r = 16″ and L = 12″,

therefore 16· = 12 or that

= 12/16 = ¾ rad

b. A slice of pizza cut from a pizza with 32” diameter

has a crust measured 12”. Find its central angle.

Give the answer in radian and degree.

Example A.

a. A slice of pizza with central angle of 50o is cut from

a 36”-diamter pizza, what is the length of its crust?

The point here is that we need radian!

50o = 50*(π/180 rad) = 5π/18 rad, with r = 18,

the length of the crust is r = 18·

5π

18 = 5π 15.7″

50o

18

?

16

?

12

a. b.

Arc Length Formula

Given that r = 16″ and L = 12″,

therefore 16· = 12 or that

= 12/16 = ¾ rad

b. A slice of pizza cut from a pizza with 32” diameter

has a crust measured 12”. Find its central angle.

Give the answer in radian and degree.

180o

πin degree, = ¾ rad = ¾ *

Example A.

a. A slice of pizza with central angle of 50o is cut from

a 36”-diamter pizza, what is the length of its crust?

The point here is that we need radian!

50o = 50*(π/180 rad) = 5π/18 rad, with r = 18,

the length of the crust is r = 18·

5π

18 = 5π 15.7″

50o

18

?

16

?

12

a. b.

Arc Length Formula

Given that r = 16″ and L = 12″,

therefore 16· = 12 or that

= 12/16 = ¾ rad

b. A slice of pizza cut from a pizza with 32” diameter

has a crust measured 12”. Find its central angle.

Give the answer in radian and degree.

180o

π

135o

πin degree, = ¾ rad = ¾ * = ≈ 43.0o

Example A.

a. A slice of pizza with central angle of 50o is cut from

a 36”-diamter pizza, what is the length of its crust?

The point here is that we need radian!

50o = 50*(π/180 rad) = 5π/18 rad, with r = 18,

the length of the crust is r = 18·

5π

18 = 5π 15.7″

50o

18

?

16

?

12

a. b.

r

Area Formula

Radian Area Formula

Given a circle of radius= r,

and a central angle = in radian,

the area A of the slice cut out

by is r2/2, i.e. A = r21

2

r

A = area Area Formula

Radian Area Formula

Given a circle of radius= r,

and a central angle = in radian,

the area A of the slice cut out

by is r2/2, i.e. A = r21

2

A= r21

2

r

A = area Area Formula

Radian Area Formula

Given a circle of radius= r,

and a central angle = in radian,

the area A of the slice cut out

by is r2/2, i.e. A = r21

2

A= r21

2

Example B. a. A slice of pizza with central angle of

50o is cut from a 36”-diamter pizza, what is the area

of the slice?

r

A = area Area Formula

Radian Area Formula

Given a circle of radius= r,

and a central angle = in radian,

the area A of the slice cut out

by is r2/2, i.e. A = r21

2

A= r21

2

Example B. a. A slice of pizza with central angle of

50o is cut from a 36”-diamter pizza, what is the area

of the slice?

r

A = area

Converting degree to radian

50o = 5π/18 rad,

Area Formula

Radian Area Formula

Given a circle of radius= r,

and a central angle = in radian,

the area A of the slice cut out

by is r2/2, i.e. A = r21

2

A= r21

2

Example B. a. A slice of pizza with central angle of

50o is cut from a 36”-diamter pizza, what is the area

of the slice?

r

A = area

Converting degree to radian

50o = 5π/18 rad, with r = 18,

the area of the slice is

5π

18

Area Formula

Radian Area Formula

Given a circle of radius= r,

and a central angle = in radian,

the area A of the slice cut out

by is r2/2, i.e. A = r21

2

1

2

A= r21

2

r2= *182*

1

2

Example B. a. A slice of pizza with central angle of

50o is cut from a 36”-diamter pizza, what is the area

of the slice?

r

A = area

Converting degree to radian

50o = 5π/18 rad, with r = 18,

the area of the slice is

5π

18

Area Formula

Radian Area Formula

Given a circle of radius= r,

and a central angle = in radian,

the area A of the slice cut out

by is r2/2, i.e. A = r21

2

1

2

A= r21

2

r2= *182*

1

2

Example B. a. A slice of pizza with central angle of

50o is cut from a 36”-diamter pizza, what is the area

of the slice?

= 45π 141 in2

Area and Arc Length

Example C.

A 24-inch2 slice of pizza is cut from a

16-inch diameter pizza, what is the length of its crust?

Area and Arc Length

Example C.

A 24-inch2 slice of pizza is cut from a

16-inch diameter pizza, what is the length of its crust?

Area and Arc Length

Example C.

A 24-inch2 slice of pizza is cut from a

16-inch diameter pizza, what is the length of its crust?

Need the angle .

Area and Arc Length

Example C.

A 24-inch2 slice of pizza is cut from a

16-inch diameter pizza, what is the length of its crust?

Need the angle . A = 24, r = 8, and A = r2/2

hence 24 = 82/2 = 64/2

Area and Arc Length

Example C.

A 24-inch2 slice of pizza is cut from a

16-inch diameter pizza, what is the length of its crust?

Need the angle . A = 24, r = 8, and A = r2/2

hence 24 = 82/2 = 64/2

24 = 32 ¾ rad =

Area and Arc Length

Example C.

A 24-inch2 slice of pizza is cut from a

16-inch diameter pizza, what is the length of its crust?

Need the angle . A = 24, r = 8, and A = r2/2

hence 24 = 82/2 = 64/2

24 = 32 ¾ rad =

Therefore L = r = 8*¾ = 6″

Area and Arc Length

Example C.

A 24-inch2 slice of pizza is cut from a

16-inch diameter pizza, what is the length of its crust?

Need the angle . A = 24, r = 8, and A = r2/2

hence 24 = 82/2 = 64/2

24 = 32 ¾ rad =

Therefore L = r = 8*¾ = 6″

Example D.

A slice of pizza with 12-inch crust is cut from a

16-inch diameter pizza, what is its area?

Area and Arc Length

Example C.

A 24-inch2 slice of pizza is cut from a

16-inch diameter pizza, what is the length of its crust?

Need the angle . A = 24, r = 8, and A = r2/2

hence 24 = 82/2 = 64/2

24 = 32 ¾ rad =

Therefore L = r = 8*¾ = 6″

Example D.

A slice of pizza with 12-inch crust is cut from a

16-inch diameter pizza, what is its area?

The arc length L = 12, r = 8, and L = r,

hence12 = 8

Area and Arc Length

Example C.

A 24-inch2 slice of pizza is cut from a

16-inch diameter pizza, what is the length of its crust?

Need the angle . A = 24, r = 8, and A = r2/2

hence 24 = 82/2 = 64/2

24 = 32 ¾ rad =

Therefore L = r = 8*¾ = 6″

Example D.

A slice of pizza with 12-inch crust is cut from a

16-inch diameter pizza, what is its area?

The arc length L = 12, r = 8, and L = r,

hence12 = 8 or 3/2 rad =

Area and Arc Length

Example C.

A 24-inch2 slice of pizza is cut from a

16-inch diameter pizza, what is the length of its crust?

Need the angle . A = 24, r = 8, and A = r2/2

hence 24 = 82/2 = 64/2

24 = 32 ¾ rad =

Therefore L = r = 8*¾ = 6″

Example D.

A slice of pizza with 12-inch crust is cut from a

16-inch diameter pizza, what is its area?

The arc length L = 12, r = 8, and L = r,

hence12 = 8 or 3/2 rad =

So A = r2/2 = 82* * = 48 in21

2

3

2

The simplicity of these formulas is carried over to other

formulas concerning rotations.

Angular Velocity

The simplicity of these formulas is carried over to other

formulas concerning rotations. For example,

the angular velocity w of a rotating wheel is the

amount of angle rotated in one unit of time.

Angular Velocity

The simplicity of these formulas is carried over to other

formulas concerning rotations. For example,

the angular velocity w of a rotating wheel is the

amount of angle rotated in one unit of time.

The angular velocity w = (π/2) rad/sec means

the wheel rotates ¼ of a round (circle) every second.

Angular Velocity

The simplicity of these formulas is carried over to other

formulas concerning rotations. For example,

the angular velocity w of a rotating wheel is the

amount of angle rotated in one unit of time.

The angular velocity w = (π/2) rad/sec means

the wheel rotates ¼ of a round (circle) every second.

The angular velocity

w = (π/2) / sec

Assuming t is in second and w is in radian,

then the blue dot travels the arc length of

w*r every second.

Angular Velocity

r

The simplicity of these formulas is carried over to other

formulas concerning rotations. For example,

the angular velocity w of a rotating wheel is the

amount of angle rotated in one unit of time.

The angular velocity w = (π/2) rad/sec means

the wheel rotates ¼ of a round (circle) every second.

The angular velocity

w = (π/2) / sec

Assuming t is in second and w is in radian,

then the blue dot travels the arc length of

w*r every second. So in t seconds,

the linear distance D or the distance the

wheel traveled on the ground is

D = w*r*t

Angular Velocity

D=w*r*t

r

The simplicity of these formulas is carried over to other

formulas concerning rotations. For example,

the angular velocity w of a rotating wheel is the

amount of angle rotated in one unit of time.

The angular velocity w = (π/2) rad/sec means

the wheel rotates ¼ of a round (circle) every second.

The angular velocity

w = (π/2) / sec

Assuming t is in second and w is in radian,

then the blue dot travels the arc length of

w*r every second. So in t seconds,

the linear distance D or the distance the

wheel traveled on the ground is

D = w*r*t

and the dial have swiped over an area of

A = ½ w*r2*t.

Angular Velocity

D=w*r*t

A =½ w*r2 *t

r

Example D. A sphere with radius r = 5 meters is

spinning with the angular velocity w = 4π rad/sec.

a. What is its linear speed? How much distance does

a point travel along the equator in one minute?

Angular Velocity

5

w = 4π

rad/sec

Example D. A sphere with radius r = 5 meters is

spinning with the angular velocity w = 4π rad/sec.

a. What is its linear speed? How much distance does

a point travel along the equator in one minute?

Angular Velocity

5

w = 4π

rad/sec

Its linear speed is 4π(5) = 20π m/sec

Example D. A sphere with radius r = 5 meters is

spinning with the angular velocity w = 4π rad/sec.

a. What is its linear speed? How much distance does

a point travel along the equator in one minute?

Angular Velocity

5

w = 4π

rad/sec

Its linear speed is 4π(5) = 20π m/sec

There are 60 seconds in one minutes so the distance

it traveled is D = w*r*t = 4π(5)(60) = 1200π m.

Example D. A sphere with radius r = 5 meters is

spinning with the angular velocity w = 4π rad/sec.

a. What is its linear speed? How much distance does

a point travel along the equator in one minute?

b. How much distance does the point p on

the sphere as shown travel in one minute?

What is its linear speed?

60o

p

Angular Velocity

Its linear speed is 4π(5) = 20π m/sec

There are 60 seconds in one minutes so the distance

it traveled is D = w*r*t = 4π(5)(60) = 1200π m.

5

w = 4π

rad/sec

Example D. A sphere with radius r = 5 meters is

spinning with the angular velocity w = 4π rad/sec.

a. What is its linear speed? How much distance does

a point travel along the equator in one minute?

b. How much distance does the point p on

the sphere as shown travel in one minute?

What is its linear speed?

60o

p

Angular Velocity

Its linear speed is 4π(5) = 20π m/sec

There are 60 seconds in one minutes so the distance

it traveled is D = w*r*t = 4π(5)(60) = 1200π m.

5

r

The radius of the rotation is

r = 5 sin(30o) = 5/2 meters.

w = 4π

rad/sec

Example D. A sphere with radius r = 5 meters is

spinning with the angular velocity w = 4π rad/sec.

a. What is its linear speed? How much distance does

a point travel along the equator in one minute?

b. How much distance does the point p on

the sphere as shown travel in one minute?

What is its linear speed?

60o

p

Angular Velocity

Its linear speed is 4π(5) = 20π m/sec

There are 60 seconds in one minutes so the distance

it traveled is D = w*r*t = 4π(5)(60) = 1200π m.

5

r

The radius of the rotation is

r = 5 sin(30o) = 5/2 meters.

So p travels 600π at a linear speed of

10π m/sec.

w = 4π

rad/sec

Exercise

Radian–Measurement Applications

At Pizza Grande, a medium pizza has 12–inch

diameter and a large pizza has 18–inch diameter.

A large pizza is cut into 8 slices sold at $3/slice and a

medium one is cut into 6 slices and sold at $2/slice.

1. Find the perimeter and the area of a medium slice.

2. Find the perimeter and the area of a large slice.

3. Which is a better deal, a medium or a large slice?

4. We want to cut one slice from the medium pizza

that is the size of two large slices. What is the central

angle of the medium–slice?

5. We want to cut one slice from the large pizza that

is the size as three medium slices. What is the

central angle of the large–slice?

6. A 25 in2 slice of pizza has a 8″ crust.

How much is the rest of the pizza

7. A 25 in2 slice of pizza has a 35o central angle.

How much is the rest of the pizza

8. A slice of pizza has a 8″ crust and a 35o

central angle. How much is the rest of the pizza

9. A slice of pizza cut from a pizza with 9″ radius

has a 8″ crust. What is the central angle of the slice?

How much is the rest of the pizza

Radian–Measurement Applications

10. A car has 18”–radius wheels is traveling with the

angular velocity of w = 10π rad/sec. How fast is car

traveling in mph?

Radian–Measurement Applications

11. A car has 15”–radius wheels, what is the approx.

angular velocity (rad/sec) of the wheels when it’s

traveling at a speed of 60 mph?

12. A radar spins at rate of w = π/4 rad/sec and has a

20–mile radius effective detection area.

In 3 second, how much area is scanned by the radar?

13. From problem 12, how long would it take for the

radar to scanned an area of 100 mi2?

≈ 8000 mi

Tropic

of Cancer

Arctic

Circle

≈ 23o ≈ 66o

14. Following are approximate

measurements of earth.

Find the linear speeds in mph

at the equator, at the Tropic of

Cancer and at the Arctic Circle.

Radian–Measurement Applications

11. A car has 15”–radius wheels, what is the approx.

angular velocity (rad/sec) of the wheels when it’s

traveling at a speed of 60 mph?

12. A radar spins at rate of w = π/4 rad/sec and has a

200–mile radius effective detection area.

In 3 second, how much area is scanned by the radar?

13. From problem 11, how long would it take for the

radar to scanned an area of 100 mi2?

≈ 8000 mi

Tropic

of Cancer

Arctic

Circle

≈ 23o ≈ 66o

14. Following are approximate

measurements of earth.

Find the linear speeds in mph

at the equator, at the Tropic of

Cancer and at the Arctic Circle.