Arclength
Consider a circle of radius r and an angle of x degrees as shown in the …gure below. The segment of the
circle opposite the angle x is called the arc subtended by x. We need a formula for its length. It is easily
obtained using a "proportion" argument:
r
arc
x
If x = 180 then the arc subtended by x is a semicircle which we know has length r. Now we argue that:
A 180 angle subtends an arc of length
r
r
180
2 r
r
=
Therefore a 2 angle subtends an arc of length 2
180
180
r
3 r
Likewise, a 3 angle subtends an arc of length 3
=
180
180
It follows that a 1 angle subtends an arc of length
In general, an angle of x degrees subtends an arc of length x
r
x r
=
180
180
We record this as a special result:
In a circle of radius r, an angle of x degrees subtends an arc of length
Example 1 In a circle of radius 25 inches, an angle of 92 subtends an arc of length
This is close to 40.1 inches.
Example 2 In a circle of radius 4.9 meters, an angle of 148 subtends an arc of length
This is close to 12.7 meters.
x r
180
(92) (25)
inches.
180
(148) (4:9)
meters.
180
Example 3 In a circle of radius 5:87 feet, an angle of 25 360 4800 subtends an arc of
25 +
36
60
48
+ 3600
180
(5:87)
f eet:
Rounded to 2 decimal places, the length is 2:62 feet. Note that we had to convert 25 360 4800 into degrees
because the formula gives the length when the angle is in degrees.
Exercise 4 Calculate the length of the arc subtended by the given angle x on a circle with the given radius
r.
x = 115:3 , r = 25 cm
x = 24 480 , r = 108 feet x = 3 120 4500 , r = 3960 miles
1
Practice problems
1. An angle of 37 subtends an arc of length 23 cm in some circle. What is the radius of the circle?
2. What angle, in degrees, minutes and seconds, (rounded to the nearest second), subtends an arc of
length 40 feet in a circle of radius 18 feet?
3. What is the angle that the minutes hand of a clock turns through between 8:15 am and 8:57 am?
4. Find the length of the arc subtended by an angle of 175 240 4500 in a circle of radius 8 yards.
5. What is the angle that the hour hand of a clock turns through between 7:00 am and 9:40 am.?
6. A clock has a minute hand that is 2.4 feet long. How many feet does its tip move between 9:43 am
and 11:21 am?
7. (This is an old tricky problem.) At 12:00 noon, the hour hand and the minute hand of a clock point
at the same spot. They NEXT point at the same spot, (after the minute hand has gone around once),
some time after 1:05 pm. What is the exact time?
8. The …gure shows a Norman window. The section WXY is a semi-circle and the section YUVW is part
of a rectangle. If YU has length 3.5 feet and UV has length 3 feet, what is the total distance around
the window?
X
W
Y
:
U
V
9. A running track consists of an inner rectangle of length 100 yards and width 50 yards, plus semicircular
legs at both ends as shown in the …gure below. There are 4 lanes and each lane is 1 yard wide.
_ _ _ _
50 yards
100 yards
_ _ _ _
2
(a) A runner does one lap in the inner-most lane. What is the total distance she covers? (You may
assume that she runs along the innermost edge of her lane.)
(b) What head-start should a runner in the outermost lane be given so that she covers the same
distance, in one lap, as the runner in the inner-most lane? (You may also assume that she runs
along the innermost edge of her lane.)
Area of a sector
The …gure below shows a circle of radius r units and an angle of x degrees. The shaded part of the circle is
called the sector subtended by the angle x.
This time the question to address is: What is the area of the sector in terms of x and r? We follow the
very route we took to …nd the length of a segment subtended by an angle of x degrees. The area of a circle
with radius r is r2 square units. It follows that the area of the sector subtended by an angle of 180 must
be one half of r2 which is 12 r2 square units. In other words:
An angle of 180 degrees subtends a sector with area 21 r2 square units. Therefore:
A 1 angle subtends a sector with area
r2
360
A 2 angle subtends a sector with area 2
A 3 angle subtends a sector with area 3
r2
360
r2
360
2 r2
360
3 r2
=
360
=
In general, an angle of x degrees subtends a sector with area x
r2
360
=
x r2
360
For future use we record:
x r2
360
In a circle of radius r units, an angle of x degrees subtends a sector with area
2
Example 5 In a circle of radius 16 cm, the sector subtended by an angle of 226 has area
centimeters. This is approximately equal to 504: 9 square centimeters.
(16) 226
square
360
Exercise 6 You are given a circle of radius r and an angle . Calculate the area of the sector subtended by
the angle.
r = 2:5 cm, x = 14:3
r = 18 feet, x = 24 360
3
r = 3950 miles, x = 3 240 5400
The formula is also simpler if the angle is measured in radians. It should be routine to conclude that:
In a circle of radius r units, an angle of x radians subtends a sector with area
xr2
square units.
2
Exercise 7
1. Find the area of the sector subtended by an angle of the given measure, in degrees, in a circle with the
given radius. In each case, draw a diagram.
Radius of circle
Measure of angle
5 inches
332
5 centimeters
23:7
4000 kilometers
0:14
Area of segment
2. A car window wiper blade consists of a metal holder 18 inches long to which a 15 inch rubber wiper is
attached as shown in the …gure below. If it rotates through 110 , how much area does it wipe in one
sweep?
Rubber wiper
3. A vender sells two sizes of pizza by the slice. The small slice is 16 of a circular 18-inch-diameter pizza,
and it sells for $2.00. The large slice is 81 of a circular 26-inch-diameter pizza, and it sells for $3.00.
Which slice provides more pizza per dollar?
Angular and Linear Speeds
Angular Speed of a Rotating Object
Angular speeds arise where there are rotating objects like car tires, engine parts, computer discs, etc. When
you drive a car, a gauge on the dashboard, (called a tachometer), continuously displays how fast your engine,
(actually the crankshaft in your engine), is rotating. The number it displays at a particular time is called the
angular speed of the engine at that instant, in revolutions per minute, (abbreviated to rpm). For example,
an angular speed of 3000 rpm means that the engine rotates 3000 times in one minute. Since there isn’t
much space on the dashboard, 3000 rpm is actually displayed as 3 and you are instructed to multiply it by
1000, (there is symbol 1000 in the center of the tachometer).
4
For objects like the seconds hand of a clock, the minute hands of a clock, etc, that rotate relatively slowly,
the angular speed is often given in degrees or radians per unit time. For example, the angular speed of the
seconds hand of a clock may be given as 360 per minute, or 6 per second, (divide 3600 by 60), or 21600
per hour, (multiply 360 by 60).
In general, to determine the angular speed of a rotating object per unit time do the following: (i) Fix a
time interval, (ii) Record the amount of rotation in that time period, (iii) Divide the amount of turning by
the length of the time interval.
Exercise 8
1. What is the angular speed of the hour hand of a clock in:
a. Degrees per hour
b. Degrees per minute
c. degrees second
d. Revolutions per minute
2. Give the angular speed of the earth in:
a. Degrees per hour
b. Degrees per second
c. degrees per minute
Linear Speed of a Rotating Object
The tires of a vehicle must rotate in order for the vehicle to move. When they rotate clockwise, it moves
forward. It moves back when they rotate counter-clockwise. Take a typical car with tires that have radius
1 foot each. Imagine the tires making one full rotation clockwise. The …gure below shows the position of a
tire at the start, then after a half rotation and …nally after a full rotation.
R
P
R
Q
Q
_________3.14 feet____________
Q
_________3.14 feet_____________
P
R
At start
P
After a half rotation
After a full rotation
(1) (180)
The car has moved forward
= ' 3:14 feet after the …rst half rotation. It moves another 3.14
180
feet after the second half rotation. Therefore it moves a total of 2 ' 6:28 feet when the tire makes a full
rotation. Complete the table. In case you have forgotten, 5280 feet make a mile.
Number of
rotations
Distance in feet
moved by car
1
2
2 '
6:28
4 '
12:56
7
12.2
x
28
42
150 feet
1 mile
To get an idea of a linear speed, assume that the tires rotate 6 times every second. Then the vehicle moves
6 (2 ) = 12 feet per second. This is close to 38 feet per second and it is called the linear speed of the tires,
(and the vehicle) in feet per second. We may transform it into a speed per hour as follows:
Since one hour is equal to 3600 seconds, the tires move 3600
5280 feet. Therefore the linear speed of the tires is
3600 12
5280
12 feet in one hour. But 1 mile equals
= 25:7 miles per hour
5
In general, the linear speed of a rotating circular object may be calculated as follows: Imagine a tire that
has the same radius as the rotating circular object. Fix a time interval t and determine the distance s that
the tire moves in that time. Now divide s by t. The result is the linear speed of the object.
Example 9 A Farris wheel in an amusement park has radius 25 feet and it makes a full rotation every 1.6
minutes.
What is the linear speed of point on the rim of the wheel in (i) feet per second, (ii) yards per minute?
Solution: If the wheel were to roll along the ground, it would move 25 (2 ) feet in 1.6 minutes. (We have
chosen a time interval of 1.6 minutes since we know how far it rolls in that time.) Therefore its linear
speed is
50
feet per minute.
1:6
(i) Since 60 seconds make 1 minute, the linear speed in feet per second is
second
(ii) Since 3 feet make 1 yard, the linear speed in yards per minute is
50
1:6
50
1:6
60 ' 1:6 feet per
3 ' 32:7 yards per minute
Exercise 10
1. The radius of the earth is about 3960 miles and there are 5280 feet in a mile. The earth makes a full
revolution in 24 hours. Calculate the linear speed of a point on the equator in:
(a) Miles per hour.
(b) Miles per minute.
(c) Feet per second.
2. Each tire of a truck has radius 1.8 feet. Assume that they make 10 complete revolutions per second.
Calculate:
(a) The angular speed of the tires in degrees per second.
(b) The linear speed of the truck in feet per second.
(c) The linear speed of the truck in miles per hour.
3. Each tire of a certain car has radius 1.2 feet. At what angular speed, in revolutions per second, are the
tires turning when it is moving at a constant speed of 70 miles per hour? (The angular speed of the
tires may be di¤ erent from the angular speed of the engine because of the gears.)
6
Angular Speed and Linear Speed Formulas
The angular speed of an object that rotates at a constant rate is given by the formula
Angular Speed =
Angle swept out
Time taken to sweep out the angle
A conventional symbol for angular speed is the greek letter !, (pronounced "omega"). Therefore if a rotating
object sweeps out an angle of x degrees in some speci…ed time t, (this may be in seconds, minutes, hours, or
other units of time), then its angular speed is
!=
x
degrees per unit time:
t
Example 11 Wanda noticed that when the ceiling fan in her room is set to operate at "very low" speed, it
makes a full rotation in 5 seconds.
This means that a point of the fan sweeps out an angle of 360 in 5 seconds. Therefore its angular speed is
!=
360
= 72 degrees per second
5
The linear speed of an object that is moving in a circle at a constant rate is the distance it travels per
unit of time. To calculate it, one measures the distance s it travels along the circular path in a speci…ed
length of time t then divided s by t. A conventional symbol for speed is v. Therefore
v=
s
t
Example 12 Say a point at the end of a blade in the above fan is 26 inches from the center of rotation of
the fan. Then in 5 seconds, such a point moves 2 (26) = 52 inches. Therefore its linear speed is
52
5
inches per second.
Rounded o¤ to 1 decimal place, this is 32:7 inches per second.
Relation Between Angular Speed and Linear Speed
When a car is moving fast, its tires are rotating fast, and vice versa. Therefore angular speed and linear
speed must be related by some equation. To determine such an equation, consider a car tire that is rotating.
Say it has radius r feet and it rotates through x degrees in a period of t seconds. Then in those t seconds,
rx
the car travels a distance of 180
feet. Therefore its linear speed is
v=
rx
feet per second.
180t
7
Its angular speed is
!=
x
degrees per second
t
r x
Notice that we may write the linear speed as v = 180
feet per second. If we replace
t
conclude that the linear speed is
r!
v=
feet per second.
180
This is the required relation, if the angles are measured in degrees.
x
t
by ! we
Review Problems:
1. Convert 49 390 4800 into decimal degrees and round o¤ your answer to 3 decimal places.
2. Convert 250.53 into degrees, minutes and seconds and round o¤ to the nearest second.
3. An angle of 48 subtends an arc of length 25 cm. in a given circle. What is the radius of the circle?
4. Calculate the area of the sector subtended by an angle of 248 in a circle of radius 5.8 feet.
5. An angle of 78:6 subtends a sector with area 124.5 square cm. in a given circle. What is the radius
of the circle?
6. An object moves along a circle of radius 10 feet and it sweeps out an angle of 60 per second. Calculate
its linear speed in (a) feet per second, (b) miles per hour.
7. How many inches does the tip of the minute hand of a clock move in 1 hour and 25 minutes if the hand
is 2 inches long?
8. Find the length of an arc subtended by an angle of 98 500 2500 in a circle of radius 12 feet.
9. An object moves along a circle of radius 30 feet and it sweeps out an angle of
Calculate its linear speed in (a) meters per minute, (b) kilometers per hour.
9
radians per minute.
2
10. Find the linear speed in miles per hour of the tire with radius 14 inches and rotating at 850 revolutions
per minute. (The angular speed of a car tire may be di¤erent from the engine speed of the car because
of gears.)
11. What is the area, rounded to 3 decimal places, swept out by a 9 inch minute hand of a clock between
8:10 am and 8:55 am?
12. There are two slices of pizza: one is 16 of a circular pizza with radius 14 inches and it costs $3.00. The
other one is 51 of a circular pizza with radius 12 inches and it costs $2:50. Which of the two provides
more pizza per dollar? Show how you arrive at your answer.
8
Radian Measure of Angles
A larger unit than a degree is called a radian. It is the angle that subtends an arc of length equal to the
radius of a circle. It is shown in the …gure below.
Exercise 13 Draw an angle of:
1. 2 radians by measuring o¤ an arc of length 2r:
2. 3.5 radians by measuring o¤ an arc of length 3:5r.
3.
2 radians.
4.
2:5 radians.
To get the relation between degrees and angles, we read the above de…nition of a radian "backwards":
In a circle of radius r; an arc of of length r is subtended by an angle of 1 radian
It follows that
In a circle of radius r; an arc of of length 2r is subtended by an angle of 2 radians
In a circle of radius r; an arc of of length 3r is subtended by an angle of 3 radians
In a circle of radius r; an arc of of length 2 r is subtended by an angle of 2 radians
Because the circumference of a circle with radius r is 2 r, the sentence in bold letters tells us that the
circumference of a circle subtends an angle of 2 radians. But it also subtends an angle of 360 . It follows
that 2 radians equal 360 .Therefore
1 radian equals
360
180
=
2
degrees
This is a little over 57:3 .
We record it for future use:
1 radian =
360
180
=
' 57:3 degrees
2
9
(1)
This conversion factor enables us to change from radians into degrees. For example,
2 radians
=
3 radians
=
180
360
2=
540
3:56 radians
=
x radians
=
degrees.
degrees
640: 8
180x
degrees
degrees
We record the formula for x radians as a special result:
x radians are equal to
180x
degrees.
The common angles in radians (the equivalent measure in degrees are also included) are:
Radians
0
Degrees
0
6
4
3
2
2
3
3
4
5
6
30
45
60
90
120
135
150
180
Radians
7
6
5
4
4
3
3
2
5
3
7
4
11
6
2
Degrees
210
225
240
270
300
315
330
360
We can also easily change from degrees into radians, by simply working backwards. More precisely, since
180
degrees are equal to one radian, it follows that 1 degree equals
radians. We also record this for
180
later use:
1 degree =
radian.
(2)
180
Using this, we conclude that:
2 degrees
=
2
radians.
180
3 degrees
=
3
radians.
180
4:7 degrees
=
4:7
radians.
180
x degrees
=
x
radians.
180
10
Arclengths And Areas When Angles Are Measured In Radians
One advantage a radian measure has over a degree measure is that we calculate arc-lengths and areas more
easily when angles are measured in radians. To see how simple, note that in a circle of radius r units, an
angle of 1 radian subtends an arc of length r units. It follows that:
an angle of 2 radian subtends an arc of length 2r units,
an angle of 3 radian subtends an arc of length 3r units,
an angle of 4.8 radian subtends an arc of length 4:8r units,
an angle of x radians subtends an arc of length rx units.
We record the last statement for future use:
In a circle of radius r units an angle of x radians subtends an arc of length rx units
Likewise in a circle of radius r, an angle of 2 radians, (which is 360 ), subtends a sector with area r2 .
It follows that:
an angle of 1 radian subtends a sector with area
r2
r2
=
square units
2
2
an angle of 2 radians subtends a sector with area
2r2
square units
2
an angle of 3.7 radians subtends a sector with area
an angle of x radians subtends a sector with area
3:7r2
square units
2
xr2
square units
2
We also record the last statement for future use:
In a circle of radius r units, an angle of x radians subtends a sector with area
xr2
square units.
2
Example 14 In a circle of radius 28 cm, an angle of 2.5 radians subtends an arc of length 28
cm.
Example 15 In a circle of radius 3 ft, an arc of length 6.9 ft is subtended by an angle of
2:5 = 70:0
6:9
= 2:3 radians.
3
Example 16 If an arc of length 66 meters is subtended by an angle of 4.8 radians then the circle must have
66
radius
= 13: 75 meters.
4:8
Exercise 17
1. Convert each angle, (given in degrees), into radians. When necessary, round o¤ your answer to 3
decimal places.
210
300
330
225
109
255.4
248
147 510 2400
2. Convert each angle, (given in radians), into degrees. When necessary, round o¤ your answer to 2
decimal places
2:3 radians
15
radians
3
7
11
6
radians
radians
11
5.53 radians
2:21 radians
3. Find the length of the arc subtended by an angle of the given measure, in radians, in a circle with the
given radius. In each case, draw a diagram.
Radius of circle
Measure of angle
4.9 meters
3.77 radians
1.6 yards
4.8 radians
3000 kilometers
0.041 radians
12
Length of arc