Lesson Title:
Radius, Radians and Pi
Date: _________________________
Teacher(s): Livas
Course:
Algebra2
Start/end times: _________________________
Lesson Objective(s): What mathematical skills and understanding will be developed?
F.TF.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the
angle.
Essential Skill and Understanding: Knowledge that angle measures in radians may be determined by a
ratio of intercepted arc to radius.
SWBAT understand the relationship between the radius of a circle, the measure of its central angle in
radians, its area, and the constant pi.
Lesson Tasks, Problems, and Activities (attach resource sheets): What specific questions, problems,
tasks, investigations, or activities will students be working on during the lesson?
Students will be given a handout with diagrams of 5 different circles and directions for how to obtain data
for each circle as well as a table for recording the data. Two columns in the table require calculated values.
1. Use cheerios to measure the radius and circumference and record your results
in the table below. (Lay them along a line/curve with adjacent cheerios touching.)
2. Calculate the ratio of circumference to radius and record.
3. Measure the angle subtended by an arc length of one radius
with the center of the circle (  ). Record result in the table.
- Measure the length of one radius along the circumference
of the circle, starting at the horizontal line shown and
going counter-clockwise. (Use the same number of
cheerios as the radius to measure this arc length.)
- Draw a line from the center of the circle to the point one
radius arc length away from horizontal. (See Figure 1.)
- Use a protractor to measure angle  .
4. Use cheerios to measure the area. Fit as many cheerios as possible on one layer inside the
marked circle. (See Figure 2.) Count the number of cheerios and record your results.
Circle
Radius
r
(# of
cheerios)
Circumference
C
(# of cheerios)
C
r
Subtended
Angle

(degrees)
Area
A
(# of
cheerios)
A
r2
A
B
C
D
E
HCPSS Secondary Mathematics Office; adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student achievement.
Portsmouth, NH: Heinemann.
Lesson Title:
Radius, Radians and Pi
Date: _________________________
Teacher(s): Livas
Course:
Algebra2
Start/end times: _________________________
Following the data collection page, students are asked a series of questions to help them interpret their
data.
Based on the values in your table, answer the following questions.
1. What can you conclude about the relationship between the radius of any circle and its
circumference? Explain.
2. What can you conclude about the relationship between the radius of any circle and its subtended
angle (  ) to the circle’s center point? Explain.
3. What can you conclude about the relationship between the radius of any circle and its area?
Explain.
4. How does the size of a circle affect these relationships? Justify your answer.
Evidence of Success: What exactly do I expect students to be able to do at the end of the lesson, and how will I
know? That is, deliberate consideration of what performances will convince you (and any outside observer) that
your students have accomplished your objective.
Students will have a completed table and have answered all of the questions on the handout.
Students will understand why there are 2 radians in 360.
Students will be able to convert from degrees to radians and radians to degrees.
Lesson Launch Notes: Exactly how will you use the
first five minutes of the lesson?
Lesson Closure Notes: Exactly what summary activity,
questions, and discussion will close the lesson and provide
a foreshadowing of tomorrow?
Warm-Up Question: Write down everything that
Define radian with class to standardize definition in
you know about circles.
student notes:
Make a list on the board of student answers to
“A unit of angular measure equal to the angle
access prior knowlegde.
subtended at the center of a circle by an arc equal in
length to the radius of the circle, approximately
Introduce the lab activity and distribute materials.
57°17’44.806” or 57.296.”
Radians are a unit of measure used in trigonometry
to describe the central angles of a circle. (Degrees “A unit of angle measure, equal to an angle at the
are another unit for measuring angles but degrees center of a circle whose arc is equal in length to the
are not so convenient for trigonometry and more radius.”
advanced math.) Using several different circles,
you will determine the relationship between the
radius of the circle and its circumference. Then
you will determine the relationship between the
radius of a circle and the central angle subtended
by an arc length equal to its radius (one radian).
Finally, you will investigate the area of a circle and
determine the relationship with its radius
measurement.
HCPSS Secondary Mathematics Office; adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student achievement.
Portsmouth, NH: Heinemann.
Lesson Title:
Radius, Radians and Pi
Date: _________________________
Teacher(s): Livas
Course:
Algebra2
Start/end times: _________________________
How many radians are in a circle? 2
How many radians are in a semicircle? 
What is the relationship between radians and
degrees? 2 radians = 360 or  radians = 180
Why is pi such an important value when working
with circles?
Notes and Nuances: Vocabulary, connections, common mistakes, typical misconceptions, etc.
circle, radius, circumference, subtended angle, degree, radian, area, pi
Resources: What materials or resources are essential
for students to successfully complete the lesson tasks or
activities?
Homework: Exactly what follow-up homework tasks,
problems, and/or exercises will be assigned upon the
completion of the lesson?
Handout for each student – circle relationships
Cheerios
Protractors
Calculators
Find the measure of the following angles in radians. Do
not use decimals. Leave answers with  and in fraction
form if needed.
1. 0
5. 90
9. 180 13. 270 17. 360
2. 30 6. 120 10. 210 14. 300
3. 45 7. 135 11. 225 15. 315
4. 60 8. 150 12. 240 16. 330
Handout for each student - homework
Find the measure of the following angles in degrees. You
may use decimals.
18. /5
20. /8
22. /7
24. 
19. /12
21. /10
23. 11/12 25. 
Post Lesson Reflections: What questions, connected to the lesson objectives and evidence of success, will you use
to reflect on the effectiveness of this lesson?
Were the students able to accomplish the tasks independently and complete the table?
What were the problems that students encountered when gathering data?
HCPSS Secondary Mathematics Office; adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student achievement.
Portsmouth, NH: Heinemann.
Lesson Title:
Radius, Radians and Pi
Date: _________________________
Teacher(s): Livas
Course:
Algebra2
Start/end times: _________________________
Were the students able to answer the questions?
Did the students understand the point of the lesson?
Does the lesson need to be modified?
Did the lesson meet my expectations?
HCPSS Secondary Mathematics Office; adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student achievement.
Portsmouth, NH: Heinemann.