6.1: Angles and Their Measure
Radian Measure
• Def: An angle that has its vertex at the center of a circle and intercepts an arc on the circle equal in
length to the radius of the circle has a measure of one radian.
θ
r
θ = 1 radian
r = radius of the circle
Notice that when you go one radian,
on you cover exactly the length of
the radius on the circle. So both
red portions have length r.
Note: a complete rotation in radians is 2π. (The circumference of the unit circle)
Now if we want to go from radians to degrees or from degrees to radians, we will need to use ratios to help
along with what we know about radians and degrees.
Fill in the fractions below (don’t forget to include units!),
one complete rotation in radians
=
one complete rotation in degrees
=
Ex,
Convert to radians or degrees whichever is appropriate.
1. 4◦
2. 10 rad
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Arc Length
• Picture illustrating the components of arc length
θ is the central angle
θ
r
r is the radius of the circle
The red portion is the arc (intercepted arc)
The arc length is the length of the red portion
Now lets try to figure out how to calculate the arc length given the radius and central angle. Fill in the
blanks below. Hint: It might be helpful to look at the picture in Radian Measure
• If the central angle is 1 radian and the radius is r, then the arc length (red portion) is
.
• If θ = 2 radians and the radius is r, then the arc length (red portion) is
.
• If θ = 3 radians and the radius is r, then the arc length (red portion) is
.
• If θ = 52 radians and the radius is r, then the arc length (red portion) is
.
• Arc Length formula
The length of an arc, s, intercepted on a circle of radius r by a central angle of θ radians is
s=
Question:
Does this formula work if θ is in degrees?
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Area of a Sector
Now we want to figure out how to find the area of the shaded region below.
θ is the central angle
θ
r
r is the radius of the circle
The red portion is the sector
The area of the sector is the area of the red portion
We can use proportions to figure out the area of the sector. We will compare the area to the central angle.
Fill in the blanks below (use radians),
Area of a sector
θ
=
Area of a sector
θ
=
Area of a Circle
Complete rotation
Now solve for “Area of a sector”
• Area of a Sector formula
The area of a sector, A, of a circle of radius r and central angle of θ radians is
A=
Question:
Do you get the same formula if you follow the process with θ in degrees?
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Rotation and Speed
When looking at a rotating object, there are two different types of speed to consider.
Angular Speed
• Def: The angular speed, ω, measures the speed of rotation.
θ
ω =
t
θ is the angle that is traversed
t is the amount of time that it takes to traverse the angle θ
This formula falls in line with our typical definition of speed. If it takes me 2 hours to travel 120 miles.
120 miles
= 60 mph. So if I go a “distance” of θ radians in say t
Then I traveled at a speed of roughly
2 hours
θ rad
.
minutes, then I traveled at a rate of
t min
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Linear (Tangential) Speed
• Def: The linear (tangential) speed, v, measures the speed of an object in rotation at a distance r
away from the center of rotation.
r is the distance from the object to the center of rotation
v = rw
ω is the angular speed (speed of rotation)
The linear speed can be thought of the speed of the object (the red point) if it were to “fly off” its path
along the circle and keep going (as illustrated below).
r
It might seem a bit weird that there is a difference between angular speed and linear speed. However, we
can see the difference if we think about the distance to the center. If two objects (red point) and (blue
point) are both traveling along a circular path at the same angular speed (so they cover the same angle in
the same amount of time), then one point actually has travels faster.
The blue dot has to travel faster since it has to cover a greater distance in the same amount of time! Why,
because the blue dot is farther away from the center, so the arc it travels is longer.
θ
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Examples
Here are some problems for you to try and practice some of the topics covered in this section.
1. A 16” thin crust pizza is 16” in diamter and typically cut into 10 slices.
(a) About how much crust does each slice have? Hint: Draw a picture!
(b) About how much area does each slice cover?
2. The first Ferris Wheel built had 36 cars, was 264 ft tall, and took 20 minutes to complete two
revolutions (complete rotations).
(a) How quickly does the Ferris wheel rotate?
(b) How fast was each car going?
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