Length of Arc Application

An hour hand on a clock is 15cm in length.
In 7 hours, how far does the hour hand move?
What is the angle the hand has moved in &theta and in degrees?

Solved:
12 hours on a clock (full circle, 2Π)
7 hours pass =>
(7/12)(2Π) = 7Π/6θ or 210°

Distance:
s=rθ
s=(15cm)(7Π/6θ) => 54.98cm

Relationship of Linear Velocity and Angular Velocity

v=rw
(s/t)=r(θ/t)

Angular Velocity

Angular velocity measures how fast an angle is changing.
w = angle in radians / time => θ/t

Linear Velocity

Linear velocity measures how fast an object is going.
Velocity = arc length / time:
v=s/t

Area of a Sector of a Circle

The area K of a sector (a slice of pie) of a circle with radius r and angle θ is:
k=1/2(r²θ)

*If you are given the angle in degrees, remember to convert to radians

Relationship of Circle Circumference, Length of Arc,Radians, Degrees

If you are given an angle in degrees you can find the length of the arc intercepted by this angle.

Example :
120° => (2π/3)θ

If the circle is of radius 1:
s=(1)(2π/3)θ =>
(2π/3)θ

Unit circle image

Formula for Circles

Formula of a circle:
X²+Y²=r²

Circumference of circle:
2Πr

Setting radius to 1 (one unit) results in:
X²+Y²=1²
2Π

Length of Arc

The portion of the perimeter of a circle of a section indicated by an angle θ has a length :
s = rθ

where s is the length in same units r is given

Angles in DMS(Degrees Minutes Seconds), Decimal and Radian (θ)

Degree 1/360 of the circumference of a circle.
Minute 1/60 of a degree
Second 1/60 of a minute

Angles in decimal notation can be converted to Degree, Minute, Second(DMS).
The whole number will remain the degrees, the decimal will be multiplied by 60(minutes in degree then 60 seconds in minute) :

64.8° =>
a) 64° + (0.8*60) =>
64°45'

31.37° =>
a) 31° + (0.37*60)' =>
b) 31° + (22.2)' =>
c) 31° + 22' + (0.2*60)" =>
31°22'12"

Reverse can be done to take DMS to decimal format :

79°19' =>
a) 79 + (19/60) =>
b) 79 + 0.3166... (round to tenths) =>
79.3°

59°23'17" =>
a) 59 + (23/60) + (17/3600) =>
59.39°


Examples of operations concerning numbers in DMS format:
1/4(40°24') = 10°6'

1/2(139°26') =>
a) 69.5° + 13' =>
b) 69° + (0.5*60)' + 13' =>
69°43'

1/4(71°17') =>
a) 17.75° + 4.25' =>
b) 17° + (0.75*60)' + 4' + (0.25*60)" =>>
17°49'15"

Radians (θ) (Converting between degrees and radians)

Π = 3.14159
1 radian = 180°/Π => 57.296° => 57°17'45"
1 degree = Π/180θ