Aim #14:
What is a radian and how do we measure arc length using
CC Geometry H
radians?
Do Now: 1) Find the area of the shaded region in terms of Π :
9
7.5
2) The radius of the following circle is 36 cm, and the m≮ABC = 60˚. What is the arc length
of AC?
Can an arc length have a length of 0? Why or why not?
Can an arc length have the length of the circumference, 2πr?
What is the possible range of lengths of any arc length?
______ < arc length < ______
An angle can be measured in either degrees or radians.
1 degree = 1/360 of a complete rotation.
1 radian = unit of measure of a central angle such that the
intercepted arc is equal in length to the radius.
1 cm
1 radian
1 cm
If a circle has radius 1 cm, a central angle of 1 radian intercepts an arc of
length 1 cm; a central angle of 2 radians intercepts an arc of 2 cm; a central
angle of 3 radians intercepts an arc of 3 cm.
3 cm
1 cm
cm
1r
1r
m
1 c
1 1r
1 cm
If a circle has radius 2 cm, a central angle of 1 radian intercepts an arc of length
2 cm. A central angle of 2 radians intercepts an arc of 4 cm; a central angle of 3
radians intercepts an arc of 6 cm.
2 c
2 c
m
2 cm
1 radian
m
1 radian
1 radian
2 cm
length of intercepted arc
In general, the radian measure of the central angle θ = ____________________
radius
θ=S
r
or
S=
What is π in radians?
A straight angle, with vertex at the center, has a semi-circle for its
intercepted arc. The length of a semi-circle is half the circle, or
S = ½(2πr) = πr.
0
πr
Therefore, the radian measure of a straight angle (180 ) is
π.
r =
0
πr
π radians and 180 are measures of the same angle.
π
0
180 = π radians
r
0
1 = _____ radians
To convert degrees to radians:
degrees
π
180
ex: Convert 90° to radians.
o
1 radian = _____
To convert radians to degrees:
radians
180
π
ex: Convert π to degrees.
3
1) What is the radian measure of central angle AOC?
O
2) Express an angle of 135° in radians.
4) Express an angle of 300° in radians.
3) Express an angle of
π
6
in degrees.
5) Express an angle of 2π in degrees.
9
6) Circle O has a diameter of 12. Central angle θ intercepts an arc with a length of 4π.
What is the measure of angle θ:
a) In degrees?
b) In radians?
7) The area of sector AOB in the following image is 28π.
a) Find the measurement of the central angle labeled x˚.
b) Express the central angle in radians.
8) a) Find the length of the radius.
b) Find the central angle in radians and
degrees.
8
72
9) In a circle whose radius measures 5 feet, a central angle intercepts an arc of
length 12 feet. Find the radian measure of the central angle.
10) The central angle of circle O has a measure of 4.2 radians and it intercepts
an arc whose length is 6.3 meters. What is the length, in meters, of the radius
of the circle?
11) Find the radius of circle A, as well as x, y and z.
(leave angle measures in radians and arc length in terms of π).
Note that C and D do not lie on a diameter.
Name: _____________________
CC Geometry H
Date: ____________
1. Express an angle of 80° in radians.
HW#14
2. Express an angle of 2π
3
in degrees.
3. Circle O has a diameter of 20. Central angle θ intercepts an arc with a length of 2π.
What is the measure of angle θ:
a) In radians?
b) In degrees?
4. The radius of a circle O is 8 inches. What is the length of the arc intercepted by a
central angle of the circle if the measure of the angle is 2.5 radians?
5. a. Given the concentric circles with center A and with m A = 60˚, calculate the arc
length intercepted by
A on each circle. The inner circle has a radius of 10, and each circle
has a radius 10 units greater than the previous circle.
b. An arc, again of degree measure 60˚, has an arc length of 5π cm. What is the
radius of the circle on which the arc sits?
6. The concentric circles all have center A. The measure of the central angle is
45°. The arc lengths are given. Find the radius of the inner, middle, and outer
circle. Then determine the ratio of the arc length to the radius of each circle, and
interpret its meaning.
Review:
1) In the figure, the radii of two concentric circles are 24 cm and 12 cm, mDFE =
0
120 . If chord DE of the larger circle intersects the smaller circle at C, find the
area of the shaded region in terms of π.
D
C
F
2) Find the area of the shaded region, to the
nearest hundredth, if the diameter is 32 inches.
132o
132o
3) Given circle A, find the following (round to the
nearest hundredth, if necessary).
a. m≮ CAD in degrees.
b. Area of sector CAD
E