Radians as a Unit of Measure
DATE:
1/25
CLASS PERIOD:
PreCalc
UNIT:
U.C.-1
LESSON OBJECTIVES:
Students will learn about radians through hands-on discovery. This lesson corresponds with
Common Core State Standards-Trig Functions F-TF 1. Understand radian measure of an angle as
the length of the arc on the unit circle subtended by the angle.
MATERIALS
Disks (lids from various food containers);
masking tape; instruction sheet
EVALUATION
Observations of work groups. Worksheet 1-2,
to be collected tomorrow.
HOMEWORK ASSIGNMENTS
Worksheet 1-2.
ACTIVITIES TO EXTEND UNDERSTANDING AND/OR RELATED TOPICS
Completing the entire worksheet will extend understanding.
Schedule
HW Check (10’)
Activity (30’)
Wrap-up (5’)
Activities
Check for completion of yesterday’s worksheet and go over answers,
especially the last two.
Discovering the Radian (taken from CPM Precalculus, 1.1.4).
Collect worksheets to finish in class tomorrow, or if they’re mostly done, for
grading.
.
Unit Circle 1-2
Worksheet 1-2 (Radians)
Name _____________________
Instructions for today’s activity:
1. Working with one other person, put some tape around the edge of the disk that Mr.
Tucker will give you. Wrap the tape all the way around, one complete revolution. Cut
or tear the tape right where it meets the other end, you don’t want any overlap.
Carefully unwrap the tape from the edge of the disk and lay it flat on your table. This is
your Circumference Tape.
2. Measure the length of the radius of your disk. Starting at one end of the Circumference
Tape, make a mark where one radius length ends. Continue marking at one radius
intervals until you get to the end of your Circumference Tape.
3. Count the number of radius lengths it took to get all the way around your disk. What
number did you get? ______ Did it come out to be an exact whole number? _________
4. Compare the number of radius lengths you got with the number of your partner (and
others in the class). Were their numbers the same as yours? ____________
5. Is there a relationship between a circle’s circumference and its radius? ___________
What is it?
How do you know?
6. Place your disk on the back of this sheet of paper and trace around it. (If your disk is
too big, use your partner’s disk and Circumference Tape.) Carefully mark the center (P)
of the circle on your paper. Place the Circumference Tape back on the disk. Transfer
the radius marks onto the traced circle. Label the position for the start of the tape as A
and the first radius mark as B. Draw the segments ̅̅̅̅
𝑃𝐴 and ̅̅̅̅
𝑃𝐵.
7. Let’s call the radius of the circle on the other side of this sheet one unit. What is the
̂ ? _______ You have just defined a new angle measure that is based
length of the arc 𝐴𝐵
on the length of a radius. This is more easily applied to higher mathematics than a
degree, which is loosely based on the number of days in a year. A RADIAN is nothing
more than a RADIus ANgle. In other words, if you take the length of 1 radius and lay
it on the circumference of the circle, the angle that is formed by the arc’s endpoints and
the center of the circle has a measure of 1 radian.
Worksheet 1-2 (Radians)
Page 2
Name ______________________
8. Assuming that the radius of your disk is one unit, what is its exact circumference?
9. Look again at the circle you traced on the back of the other sheet. Let the length of the
radius of this circle equal 1 unit. (A circle with a radius of one is called a unit circle.)
a. How many degrees did you go around the circle in order to mark off 2𝜋 radius
lengths (radians)?
b. A radian is a unit of measure. The central angle of a full circle is 360 degrees. How
many radians is this?
c. What whole number of radians best approximates the central angle of a full circle?
10. When we measure angles in the unit circle, we measure from standard position (the
positive x-axis) as shown in the diagram. On the diagram, show an angle with a
measure of approximately 3 radians starting from standard position.
11. We want to develop a method for comparing degrees and radians.
a. How many degrees of angle measure are there in one half of a circle? __________
b. About how many radians of angle measure are there in one half of a circle? _______
c. Exactly how many radians of angle measure are there in one half of a circle? ______
d. Write an equation that relates radians and degrees using the half circle.
e. Use this relationship to convert
𝜋
3
radians into degrees.
f. How many radians are there in 200? Give an exact value using  in your answer.
Worksheet 1-2 (Radians)
Page 3
Name ______________________
12. Convert the degree measures into radians. Use  to give an exact value.
a. 180
b. -36
c. 2
13. Convert the radian measures to degrees. Leave  in your answer if necessary to give
the exact value.
3𝜋
a.
2
7𝜋
b. −
6
c. 2
14. A wheel is spinning at 500 revolutions per minute. Remember 1 revolution equals 2
radians. How many radians per second is that?