The Polygon Angle Sum Theorem
Common Core State Standards:
G.CO.10: Prove theorems about triangles. Theorems include: measures of interior angles of a
triangle sum to 180 degrees; base angles of isosceles triangles are congruent; the segment joining
midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of
a triangle meet at a point.
G.CO.12: Make formal geometric constructions with a variety of tools and methods (compass
and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).
Before the Lesson:
Objective: Students will be able to find the sum of the measure of the interior angles of
polygons and discover the Interior Angle Sum Theorem and how it works.
Prerequisite knowledge: The students will have already learned the names of the polygons up
to 12 sides, the triangle angle sum theorem, the definitions of concave and convex polygons, the
definition of a polygon diagonal, measuring an angle with a protractor by hand and with
geobebra software, and constructing various polygons by hand and by geogebra.
Materials:
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Chromebooks
Geogebra free software
Drill
Exit ticket
Discovering the polygon Angle Sum Theorem
Projector
Document reader
Drill: (5 minutes) Students will be given 5 different regular polygons they must identify.
Students will be asked to share with the person sitting next them after 5 minutes. This is to see
what students remember during previous lesson. After students share I will go over the answers
asking for a thumbs up or thumbs down.
Lesson:
Essential Question: Is there an easier way to find the measures of the angles of a polygon
without spending extra time using a protractor?
Exploration: (15-20 mins) Instruct students to open chromebooks and open Geogebra
software. I will hand out “Discovering the Polygon Angle Sum Theorem” handout that students
will be required to complete with a partner. I will do the first two examples with them using the
projector on the board. Then students are on their own. The students will be required to measure
each angle of the polygons and find the sum of the angles within each. Students will be using the
regular polygon making tool and the angle measure tool.
Explanation: (15-20 mins) The class will reconvene after students complete discovery activity.
I will instruct them to close the Chromebooks. The same chart the students filled out will be on
the board using the document reader. I will call on students to help fill the chart out asking for
thumbs up or down. I will ask if students can spot a pattern with the results. I will then instruct
them to turn to the bottom of the Discovery hand-out where it says “Only complete with the
teacher”. There will be a series of regular polygons. I will redefine the term polygon diagonal as
a line segment connecting two non-adjacent vertices. I will also point out that the diagonals
should not intersect. Point out that we know a triangle has an angle sum measure of 180o. I will
ask the students how many triangles can be created within the various polygons. Example:
Quadrilateral: 2 x 180 = 360, Pentagon: 3 x 180 = 540, etc. Now ask students to try to create a
formula to find the sum of any regular polygon. (n – 2) x 180.
Evaluation or closure: Exit tickets. Students will be given two different regular polygons 11sided and 12-sided. Students will have to divide the polygons into triangles and then calculate
the angle sum measure using the formula (n – 2) x 180. I will walk around checking and
collecting.
Drill
Identify the following polygons by number of sides? Regular or irregular, convex or
concave?
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Discovering the Polygon Angle Sum Theorem
Construct the following polygons on Geogebra using the “regular polygon making tool”
and measure each angle using the “angle measuring tool”. Fill out the given table.
Convex Polygon
Triangle
Number of Sides
Sum of Angle Measure
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
With your partner look for a pattern in the sum of angle measures related to the number of
sides.
Only complete this portion of the activity with the teacher. (wait for instructions)
Conclusion: __________________________________
Exit Ticket
Divide the following polygons into triangles and find the angle sum measure of each
polygon using the form (n – 2) x 180.
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Exit Ticket
Divide the following polygons into triangles and find the angle sum measure of each
polygon using the form (n – 2) x 180.
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