Lesson Title: Equation of a Circle
Date: _____________ Teacher(s): ____________________
Course: Geometry (G.GPE.1)
Start/end times: _________________________
Lesson Objective(s): What mathematical skill(s) and understanding(s) will be developed?
G.GPE.1: Derive the equation of a circle of given center and radius using the Pythagorean theorem.
Lesson Launch Notes: Exactly how will you use the first five minutes
of the lesson?
1. State the Pythagorean Theorem.
2. What is it used to find?
3. Draw a right triangle with legs labeled 10 and 24. Have
students find the length of the hypotenuse.
Lesson Closure Notes: Exactly what
summary activity, questions, and discussion
will close the lesson and provide a
foreshadowing of tomorrow? List the
questions.
How do we find the equation for a circle?
What if we have the equation, can we
determine the center and radius?
Lesson Tasks, Problems, and Activities (attach resource sheets): What specific activities, investigations,
problems, questions, or tasks will students be working on during the lesson?
1. Lesson Launch – This will determine students’ prior knowledge regarding computations involving
the Pythagorean Theorem. (UDL I:3)
2. Introduction of Task – Ask students how they would define a circle; record their responses. If
necessary lead students to ultimately generate the definition similar to that in the textbook. Explain
to students that today they will be deriving the equation of a circle given the center and radius.
3. Distribute graph paper, rulers, and compasses to the class. As a whole class, instruct students to
fold the graph paper in half. Have students draw coordinate axes on each half. On the top half,
have students put the center of the circle at (0, 0) and construct a circle of radius 5. Have them plot
the point (3, 4) on the circle and then use their ruler to put in the right triangle with vertices at (0,
0), (3, 4), and (3, 0). They should label the hypotenuse r. Go through the process of finding r by
using the Pythagorean Theorem. They should be able to verify that r  5.
4. On the bottom half of the graph paper, have students once again draw a circle with radius 3 and
center at (0, 0). Have them pick a point on the circle and label it ( x, y ). Emphasize that the
equation of a circle is valid for any point on the circle, which is why the point is labeled ( x, y ).
Have students follow the same process as in the first example and construct the right triangle. They
5.
6.
7.
8.
should then use the Pythagorean Theorem to find the equation x 2  y 2  32. (UDL II:5)
In small groups, have each student construct another circle with a compass. This circle should be
centered at ( 2,3) and have a radius of 4. Allow students time to work in their groups to develop the
equation of the circle. (UDL I:3)(UDL III:8) Try another one with a center at ( 1,4) and radius
6. If time allows, challenge them to find the equation of the circle if the center is named ( h, k ) , the
point on the circle is ( x, y ), and the radius is r.
As an alternative for students that are having difficulty using the compass and graph paper, students
can use Geometer’s Sketchpad for their constructions and computations. (UDL I:1, 2, 3)(UDL
II:4, 5)
As students are working, you may want to have the following websites available:
http://www.brightstorm.com/math/trigonometry/pythagorean-theorem/equation-of-a-circle/
http://www.youtube.com/watch?v=MIfa7szV43Q&noredirect=1. Both offer visuals and guided
steps of the concepts in this lesson. (UDL I:1, 2, 3) (UDL III:7, 8)
Bring the class back together. Have groups share what they learned through their investigation.
Students should be able to provide the equation for a circle with center at ( h, k ) and radius r .
March 19, 2012
Lesson Title: Equation of a Circle
Course: Geometry (G.GPE.1)
Date: _____________ Teacher(s): ____________________
Start/end times: _________________________
Evidence of Success: What exactly do I expect students to be able to do by the end of the lesson, and how will I
measure student mastery? That is, deliberate consideration of what performances will convince you (and any outside
observer) that your students have developed a deepened (and conceptual) understanding.
Students will be able to determine the equation of any circle with a given center and radius. Have students
write an equation for the circle with center at ( 2,3) with radius 4. Have them explain why there are
addition signs in the parentheses for ( x  2) 2 ( y  3) 2  16 .
Notes and Nuances: Vocabulary, connections, common mistakes, typical misconceptions, etc.
Students must be able to apply the Pythagorean Theorem correctly.
Students may have difficulty drawing the triangle or determining the lengths of the legs once the circle is no longer
centered at ( 0,0).
Students may be confused about the signs if the center involves negative coordinates.
Resources: What materials or resources are essential for students to
Homework: Exactly what follow-up
successfully complete the lesson tasks or activities?
homework tasks, problems, and/or exercises
will be assigned upon the completion of the
lesson?
Ruler
Compass
Teachers can choose from numerous
Computers with internet access
problems in the textbook.
Geometers Sketchpad
http://www.brightstorm.com/math/trigonometry/pythagoreantheorem/equation-of-a-circle/
http://www.youtube.com/watch?v=MIfa7szV43Q&noredirect=1
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 students able to determine the equation once the center was moved from the origin? How about when the
center had negative ( h , k ) values?
Could students generalize their results to derive the equation for any circle having center at ( h , k ) and radius r ,
without using the Pythagorean Theorem?
March 19, 2012