Unit Circle, Arcs, and Sectors
Objectives:
1. To complete and use
the unit circle
2. To find
circumference and
arc length
3. To find the area of
circles and sectors
Assignment:
• Complete Unit Circle
• Challenge Problems
Objective 1
You will be
able to
complete and
use the unit
circle
Exercise 1
Solve each right triangle. Write your
answers in simplest radical form.
1 unit
1 unit
30
45
Radians
Radians are
another way to
measure an
angle. If you
take the radius
and wrap it
around the circle,
the angle that is
formed is one
radian.
Radians
It takes a little bit
more than 3
radians to span a
semicircle.
That “little bit more
than 3” is π.
So π radians = 180°
and 2π radians =
360°
Exercise 2
Rewrite each of the following angle
measures in terms of radians.
(180° = π rad)
1. 30°
2. 45°
3. 60°
4. 90°
Exercise 3
Write the equation of
the circle.
The Unit Circle
This tiny circle is
called the unit circle
since its radius is 1
unit. This circle may
be tiny, but it will give
us a way to find 102
exact trig values.
That’s pretty useful.
Unit Circle Activity
Math students often use
a unit circle to find the
exact trig ratios of
certain angle
measures, since most
calculators won’t
divulge that
information. In this
activity, we will
construct a unit circle.
y
(0, 1)
90
60
45
30
0
360
x
0
2
(1, 0)
Unit Circle Activity
The outer bold circle
is the unit circle.
We will eventually be
labeling each of the
points along this
bold circle with
ordered pairs.
y
(0, 1)
90
60
45
30
0
360
x
0
2
(1, 0)
Unit Circle Activity
Now look at the inner
circle. The points
along this circle will be
labeled with degree
measures. What do
you suppose these
degree measures
represent?
Finish the degree
measures.
y
(0, 1)
90
60
45
30
0
360
x
0
2
(1, 0)
Unit Circle Activity
Now look at the middle
circle. The points
along this circle will be
labeled with radian
measures.
Finish the radian
measures for the first
quadrant.
y
(0, 1)
90
60
45
30
0
360
x
0
2
(1, 0)
Unit Circle Activity
Finally, look at the outer
circle again. Let’s
concentrate on the
point that is 30° along
this circle.
y
(0, 1)
90
60
45
30
0
360
x
0
2
(1, 0)
Unit Circle Activity
Finally, look at the outer
circle again. Let’s
concentrate on the
point that is 30° along
this circle.
Realize that the ordered
pair for any point
makes a right triangle
with the x-axis.
y
(0, 1)
90
60
45
30
0
360
x
0
2
(1, 0)
Unit Circle Activity
1. What is the length of
the hypotenuse?
2. What is the length of
the short leg?
3. What is the length of
the longer leg?
4. What are the
coordinates of the
point on the circle?
y
(0, 1)

90
3 1
,
2 2

1
60
1
45
30
0
360
2
3
2
x
0
2
(1, 0)
Unit Circle Activity
How can this unit circle
be used to find the
following?
1. cos (30°)
2. sin (30°)
3. tan (30°)
y
(0, 1)

90
3 1
,
2 2

1
60
1
45
30
0
360
2
3
2
x
0
2
(1, 0)
Unit Circle Activity
Let’s look at the outer
circle again. This time
concentrate on the
point that is 45° along
this circle.
y
(0, 1)
90
60
45
30
0
360
x
0
2
(1, 0)
Unit Circle Activity
1. What is the length of
the hypotenuse?
2. What is the length of
the base leg?
3. What is the length of
the height leg?
4. What are the
coordinates of the
point on the circle?
y
(0, 1)

2
2
,
2
2

1
2
90
60
2
45
30
2
0 2
360
x
0
2
(1, 0)
Unit Circle Activity
How can this unit circle
be used to find the
following?
1. cos (45°)
2. sin (45°)
3. tan (45°)
y
(0, 1)

2
2
,
2
2

1
2
90
60
2
45
30
2
0 2
360
x
0
2
(1, 0)
Unit Circle Activity
In general, for any
point (x, y) along the
outer circle of the
unit circle:
1. cos( ) = x
2. sin ( ) = y
3. tan ( ) = y/x
y
(0, 1)
90
60
45
30
0
360
x
0
2
(1, 0)
Unit Circle Activity
Let’s look at the outer
circle one last time.
This time let’s look at
the point that is 150°
along the unit circle.
Obviously, we cannot
make a right triangle
with a 150° angle. So
how could we complete
the second quadrant?
y
(0, 1)
90
60
45
30
0
360
x
0
2
(1, 0)
Unit Circle Activity
The answer involves symmetry.
y
(0, 1)

-

-
2
2
3 1
,
2 2
,
2
2



2
2
,

2
2

3 1
,
2 2

90
60
45
30
0
360
x
0
2
(1, 0)
Unit Circle Activity
The same would apply for the 3rd and 4th
quadrants.
y
(0, 1)

-

-
2
2
3 1
,
2 2
,
2
2



2
2
,

2
2

3 1
,
2 2

90
60
45
30
0
360
x
0
2
(1, 0)
Objective 2
You will be able to
find circumference
and arc length
Exercise 4
A tennis ball can is
approximately 3 tennis
ball diameters high, and
its width is
approximately one
tennis ball diameter.
Which is greater: the
height of a tennis ball
can or the distance
around the can?
Let’s Have Some π
The distance around a circle is its
circumference.
Diameter
Circumference
Approximating Pi
Archimedes of Syracuse first approximated
pi as 3.14 using the perimeters of inscribed
and circumscribed polygons.
Circumference of a Circle
The circumference C of
a circle is C = πd or
C = 2πr, where d is
the diameter and r is
the radius of the
circle.
Arc Measure and Arc Length
The measure of an
arc is the measure
of the central angle
it intercepts. It is
measured in
degrees.
Arc Measure and Arc Length
An arc length is a
portion of the
circumference of a
circle. It is
measured in linear
units and can be
found using the
measure of the arc.
Exercise 5
Assume the radius of each circle below is 24
units. Find the length of arc AB.
B
B
B
90
60
A
45
A
A
Arc Length Corollary
If the radius of a circle
is 𝑟 and two radii form
a central angle of 𝑎°,
then the length of the
arc formed by those
radii is given by the
formula:
B
r
a
A
Arc Length Corollary
If the radius of a circle
is 𝑟 and two radii form
a central angle of 𝑎°,
then the length of the
arc formed by those
radii is given by the
formula:
B
r
a
A
Exercise 6
Find the length of arc AB.
1.
2.
3.
Exercise 7
Find the indicated measure.
1. Circumference =
2. Radius =
Exercise 8
Find the perimeter of the region.
1.
2.
Objective 3
You will be able to find the
area of circles and sectors
Area of a Circle
Recall that when investigating the area of certain
polygons, we based our new area formulas on
shapes whose area we already knew. We’ll do
the same with the area of a circle. (It’s probably
the coolest thing you’ve ever seen.)
Area of a Circle
Recall that when investigating the area of certain
polygons, we based our new area formulas on
shapes whose area we already knew. We’ll do
the same with the area of a circle. (It’s probably
the coolest thing you’ve ever seen.)
Area of a Circle
Recall that when investigating the area of certain
polygons, we based our new area formulas on
shapes whose area we already knew. We’ll do
the same with the area of a circle. (It’s probably
the coolest thing you’ve ever seen.)
Area of a Circle
Area of a Circle Theorem
The area of a circle is π times the square of
the radius.
Sector: An Actual Slice of Pie
A sector of a circle
is the region
between two radii
of a circle and the
included arc.
Exercise 9
Remember that delicious
cherry pie from Example
5? Well, it’s mostly a
memory now, as there is
only one slice left! What
is the area of that piece
of pie if the radius is 6
inches and the angle
formed by the sides of
the pie slice is 60°?
Got any vanilla ice
cream for that?
Area of a Sector Conjecture
If the radius of a circle
is r and two radii
form a central angle
of a°, then the area
of the sector formed
by those radii is
given by the formula
a
 r2
360
Area of a Sector Conjecture
If the radius of a circle
is r and two radii
form a central angle
of a°, then the area
of the sector formed
by those radii is
given by the formula
Exercise 10
The radius of a circle
is 18 cm. A sector
is formed by a
central angle
measuring 40°.
What is the exact
area of the sector?
Exercise 11
Find the area of each sector.
1.
2.
Exercise 12
1. Find the area of
circle S.
2. Find the radius of
circle S.
Exercise 13
What is the area of
the shaded region if
the radius of each
circle is 6 cm.
Unit Circle, Arcs, and Sectors
Objectives:
1. To complete and use
the unit circle
2. To find
circumference and
arc length
3. To find the area of
circles and sectors
Assignment:
• Complete Unit Circle
• Challenge Problems