Primary Type: Lesson Plan
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 130853
Construct Regular Polygons Inside Circles
Students will be able to demonstrate that they can construct, using the central angle method, an equilateral triangle, a square, and a regular
hexagon, inscribed inside a circle, using a compass, straightedge, and protractor. They will use worksheets to master the construction of each
polygon, one inside each of three different circles. As an extension to this lesson, if computers with GeoGebra are available, the students should be
able to perform these constructions on computers as well.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Suggested Technology: Computer for Presenter,
Interactive Whiteboard, LCD Projector, GeoGebra Free
Software (Download the Free GeoGebra Software)
Instructional Time: 50 Minute(s)
Keywords: construct, circle, polygon, equilateral triangle, hexagon, square, regular, central angle, triangle,
inscribed, GeoGebra
Resource Collection: FCR-STEMLearn Geometry
ATTACHMENTS
Square Construction.ggb
Square Final Product.ggb
Hexagon Construction.ggb
Hexagon Final Product.ggb
Summative Assessment.docx
Student Handout Key (pg1).pdf
Student Handout Key (pg2).pdf
Summative Assessment Key.pdf
Student Handout.docx
Triangle Construction.ggb
Triangle Final Product.ggb
LESSON CONTENT
Lesson Plan Template: General Lesson Plan
Learning Objectives: What should students know and be able to do as a result of this lesson?
Students will be able to construct an inscribed equilateral triangle, square, and regular hexagon inside a circle using a compass, straightedge and protractor.
Prior Knowledge: What prior knowledge should students have for this lesson?
Definition of equilateral triangle, square, and regular hexagon.
Unique properties of all regular polygons, such as: all sides congruent, all interior angles are congruent, the central angle is calculated by dividing 360 degrees by
the number of sides of the polygon.
Definition of an arc and central angle of a circle.
page 1 of 4 Guiding Questions: What are the guiding questions for this lesson?
Why can you inscribe any regular polygon inside a circle? (All interior angles are congruent, the distances from the center of the circle to all vertices of the polygon are
congruent. That distance is called the radius of the polygon and also the radius of the circumscribed circle.)
Why can you not inscribe irregular polygons inside a circle? (The radius of the circle is not always equal to the distance from the center of the circle to each vertex of
the polygon you are attempting to inscribe inside the circle.)
Teaching Phase: How will the teacher present the concept or skill to students?
Distribute a compass, straightedge, protractor, and the "Student Handout" attachment to every student.
The teacher should open the GeoGebra file called "Triangle Construction.ggb," "Square Construction.ggb," and "Hexagon Construction.ggb." The GeoGebra files have a
circle already constructed for the teacher to use and display the construction steps for students to follow using a compass, straightedge and protractor. The teacher
will perform the construction using GeoGebra while students follow along using their tools. The teacher will use GeoGebra in the guided practice.
Note to Teacher: "Practice" the constructions in GeoGebra before presenting to the students.
The "undo" and "redo" buttons are the curved arrows in the upper right corner.
Constructing a segment between points A and B: click the red arrow in the corner of the third icon on the top bar (showing a line with two points on it), and select
the second option, "Segment between two Points." Once the tool is selected, the program puts up guided instructions for the user to the right of the last icon, telling
you what to do next. In this case it says, "Segment between two points: Select two points." Click on point A then on point B, and a segment is formed between the
two points.
Constructing an angle of a given size: clicking on the red arrow on the eighth icon on the top (showing an angle), and then select the second option, "Angle with
Given Size." The prompt will show the guided instruction, "Angle with Given Size: Select Leg Point, then Vertex, and enter Size." Click the mouse on point B (on the
circle), then point A, the center or vertex, and a box opens up to enter the angle size. The teacher should enter 120, and select counter clockwise and click ok. A
120 degree angle will be formed with the center of the circle A as the vertex of the angle.
Guided Practice: What activities or exercises will the students complete with teacher guidance?
Students complete the attached Student Handout (30 minutes)
Equilateral Triangle
The teacher begins the guided practice by demonstrating the steps for constructing an equilateral triangle inside a circle. The students will then use the same steps to
construct the same inscribed triangle on their handout to complete the guided practice for constructing an equilateral triangle inside a circle.
The teacher should open and display the GeoGebra file "Triangle Construction.ggb." This GeoGebra file and a similar one for the square and hexagon are used to
display the step by step instructions for the teacher and student, which the teacher will walk through with the students as they practice the steps on their "Student
Handouts," using their compass, protractor, and straightedge. The displayed "Triangle Construction.ggb" will have the steps shown on the right for constructing a
triangle inside the circle, shown on the left. The teacher will demonstrate and talk through these steps as he constructs the triangle inside the circle. The student
should write the steps down on their "Student Handouts" as the teacher presents these steps. For reference, the teacher should print out, before the lesson, the
solution keys for the triangle, square and regular hexagon constructions from the three files, "Triangle Final Product, Square Final Product, and Hexagon Final
Product."
The teacher will help students calculate the central angle measurement for the triangle which defines each arc on the circle representing each central angle of the
triangle. The central angle will intersect the circle in two places, where marks will be made on the circle. Then the side of the triangle will be drawn as a straight line
between those two marks. The teacher will show how to copy, with a compass, this equidistant segment which has been constructed will define each congruent side
of the triangle being inscribed inside the circle.
The teacher should demonstrate the triangle construction by beginning with constructing point B on the circle, then constructing segment AB using a straightedge, and
then using the protractor to construct angle B'AB with a size of 120 degrees. The teacher will query the students to make sure they know why the central angle is 120
degrees. The teacher then completes demonstrating the steps on the screen to complete the inscribed triangle. The students will then construct, using the same
steps, on their "Student Handouts," the same inscribed triangle to complete the guided practice for constructing a triangle inside a circle. The answer key for this
portion is the file, "Triangle Final Product."
Square
To continue the guided practice, the teacher will demonstrate the steps to construct a square inside a circle, using the same central angle method used for the
triangle. The teacher should display GeoGebra file "Square Construction.ggb" on the screen. The displayed "Square Construction.ggb" will have a blank circle on the
left with the steps shown on the right for constructing a square inside the circle. The teacher will demonstrate the following steps as he constructs the square inside
the circle. Beginning with constructing point B on the circle, then constructing segment AB using a straightedge, and then using the protractor to construct angle B'AB
with a size of 90 degrees. The teacher will query the students to make sure they know why the central angle is 90 degrees. The teacher then completes
demonstrating the steps on the screen to complete the inscribed square. The students will then construct, using the same steps, on their "Student Handouts," the
same inscribed square to complete the construction. The teacher will verify that the students are now using a central angle of 90 degrees for a square. The answer
key for this portion is the file, "Square Final Product."
Regular Hexagon
To complete the guided practice the teacher will demonstrate the steps to construct a regular hexagon inside a circle, using the same central angle method used for
the square and triangle. The teacher should display GeoGebra file "Hexagon Construction.ggb" on the screen. The displayed "Hexagon Construction.ggb" will have a
blank circle on the left with the steps shown on the right for constructing a regular hexagon inside the circle. The teacher will demonstrate the following steps as he
constructs the hexagon inside the circle. Beginning with constructing point B on the circle, then constructing segment AB using a straightedge, and then using the
protractor to construct angle B'AB with a size of 60 degrees. The teacher will query the students to make sure they know why the central angle is 60 degrees. The
teacher then completes demonstrating the steps on the screen to complete the regular hexagon. The students will then construct, using the same steps, on their
page 2 of 4 "Student Handouts", the same regular hexagon to complete the construction. The teacher will verify that the students are now using a central angle of 60 degrees for
a regular hexagon. The answer key for this portion is the file, "Hexagon Final Product."
The attachment "Student Handout Key" shows the solution for checking the students' work on the "Student Handout" attachment.
Independent Practice: What activities or exercises will students complete to reinforce the concepts and skills developed in the
lesson?
The teacher will administer the summative assessment, which should be done independently. The summative assessment, which consists of three problems:
constructing an equilateral triangle, a square, and a regular hexagon inscribed in a circle, will be given to students to complete at the end of the lesson. These will be
graded to measure mastery of the learning objectives. (15 minutes)
See "Summative Assessment Key" for solutions.
Closure: How will the teacher assist students in organizing the knowledge gained in the lesson?
The students have demonstrated on paper that they can construct an equilateral triangle, a square and a regular hexagon inscribed inside a circle. The lesson will
conclude that one method to inscribe a regular polygon inside a circle is the central angle method, where the student needs to draw "n" marks equally spaced around
the circumference of the circle, where "n" is the number of sides of the polygon, and the degrees between two consecutive "n" marks is the central angle
measurement for the specific polygon being constructed.
A few students can be invited to show their "Student Handouts" in front of the classroom to display their work. They should be prompted to discuss their constructions.
The class should come to consensus about the central angle method for constructing regular polygons inside circles.
Summative Assessment
The teacher will administer the summative assessment, which should be done independently. The summative assessment, which consists of three problems:
constructing an equilateral triangle, a square, and a regular hexagon inscribed in a circle, will be given to students to complete at the end of the lesson. These will be
graded to measure mastery of the learning objectives. See "Summative Assessment Key" for solutions. (15 minutes)
Formative Assessment
During the guided practice phase, the teacher will circulate about the room offering guiding questions as needed to help students connect to the construction activity.
The teacher will take note of the progress of the students' constructions of the equilateral triangles, squares, and regular hexagons inside circles, and be available to
issue guided questions to facilitate learning. The teacher could invite several students to share their inscribed triangles, squares, and hexagons with the class during
the closure portion of the lesson.
Feedback to Students
During the guided practice portion of the lesson, the teacher circulates around the room, using guiding questions to check students' knowledge of constructing the
inscribed triangle, square, and hexagon inside circles, and to ensure students remain engaged in the lesson.
During the independent practice portion of the lesson, the teacher continues to circulate around the classroom, monitoring student progress constructing the
triangle, square, and hexagon inside the circle, and providing clarification to struggling students. During the closure phase of the lesson, some students can be
invited to share their "Student Handout" sheets with the class to verify everyone's understanding of constructing inscribed regular polygons inside circles.
The summative assessment feedback will consist of a grade for accuracy and will be returned to students.
ACCOMMODATIONS & RECOMMENDATIONS
Accommodations:
The GeoGebra software is incorporated in the lesson to assist visual learners. The GeoGebra files are preprinted in easy to read format to assist students in copying
down concise notes and instructions for the constructions. The GeoGebra files, which have the construction steps written out, may be printed ahead of time for
students. Students may work together in pairs to assist each other during the guided practice phase.
Extensions:
Students can compare the different methods of inscribing a square and regular hexagon inside a circle, using the central angle method for a regular hexagon
described in this lesson or the fact that the radius of the circle is the length of the side of a regular hexagon, and the central angle method for constructing the square
described in this lesson, or the perpendicular bisection of the diameter method for the square.
With the central angle calculation method described in this lesson, students should be able to extrapolate how this method would work to inscribe any regular polygon
inside a circle.
If computers are available with GeoGebra, the students could perform these inscribed polygon constructions on computers as well.
Plan ahead to have access to a computer lab or laptop cart so the students can use GeoGebra to practice after the paper method is attempted and mastered.
Ensure that, if computers will be used for GeoGebra that you have downloaded the latest version of GeoGebra on all computers and that they are fully functional with
full battery charge if they are laptops
Suggested Technology: Computer for Presenter, Interactive Whiteboard, LCD Projector, GeoGebra Free Software
Special Materials Needed:
Paper, pencil, provided by students
page 3 of 4 Compass, 1 per student
Straightedge, 1 per student
Protractor, 1 per student
Computers if using GeoGebra
One copy of the "Student Handout" for each student
One copy of the "Summative Assessment" for each student
Further Recommendations:
The teacher should practice the constructions using GeoGebra before the lesson until he or she is fully comfortable with the technology. Ask around at your school and
you should find an expert or two on GeoGebra. If not, the opportunity is yours to become the expert!
Additional Information/Instructions
By Author/Submitter
This lesson addresses the following Standards for Mathematical Practice:
MAFS.K12.MP.5.1 - Use appropriate tools strategically
MAFS.K12.MP.7.1 - Look for and make use of structure
MAFS.K12.MP.8.1 – Look for and express regularity in repeated reasoning
SOURCE AND ACCESS INFORMATION
Contributed by: Duane Consbruck
Name of Author/Source: Duane Consbruck
District/Organization of Contributor(s): Broward
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.912.G-CO.4.13:
Description
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
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