Triangles Everywhere

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SWBAT find the sum of the measures of the angles in any polygon and the measure of one angle inside a regular polygon.

Big Idea

Students use triangles formed by diagonals of a polygon to discover the sum of the measures of the angles inside a polygon.


5 minutes
As students enter the room, they take out their Problem of the Day (POD) sheet and begin to work on the question on the SMART Board. The POD allows students to use MP3 in the discussions we have about the problem each day.
The POD asks students to describe how triangles are related to other polygons. It is interesting to hear what connections students believe exist, and I encourage discussion. It would be exciting if a student declares that all regular polygons can be divided into triangles. Ideally, a student will have an explanation or observation that leads into a starter for our activity today. If the result of discussion is that students do not believe a relationship exists other than the fact that they are all polygons, then the revelation that results from the activity will be valuable for students. This exercise is not in an attempt to build procedural knowledge about the formulas, but rather for students to discover a connection between two concepts.


30 minutes

In this activity, students explore the measures of interior angles of polygons. By completing it, students develop a method for determining the sum of the angles in several polygons. Students begin by sectioning each polygon into triangles. This is accomplished by drawing all of the diagonals from one vertex, in other words, a line connecting one vertex with all the other non-adjacent vertices in the polygon. Since the sum of the measures of the angles in a triangle is 180°, the sum of the measures of the angles for any other polygon is equal to the product of the number of triangles that fit in it and 180°. Some students may notice that the number of triangles formed is always 2 fewer than the number of sides the polygon has, further simplifying the formula.


As a class, we use this strategy to determine a formula for the measure for each angle in a regular polygon. Remind students that angles in a regular polygon are congruent. So, students can find the sum of the measures of the angles in the polygon and divide by the number of angles. However, if a polygon is not regular, then we cannot determine with certainty the measure of any one angle without additional information.


5 minutes

To finish class, students respond to an exit ticket prompt asking them to summarize their learning in the activity. I say, “Describe how to find the sum of the measures of the angles in a pentagon.” Through this, I want to see if students understand the activity and the underlying concept. Do students get it? Do they recognize the pattern? Can they apply it to other polygons? Can they explain it?