Sum of interior Angles of a Polygon

Kei Muto, Mary-Catherine LaForest & Caitlin Reynolds

Curriculum – Measurement

  • Develop geometric relationships involving lines, triangles, and polyhedra, and solve problems involving lines and triangles
  • Solve angle-relationship problems involving triangles, intersecting lines, and parallel lines and transversals

Summary

In this lesson students will explore the relationship between the sums of interior angles in different polygons. The lesson aims to move students towards an understanding that there is a pattern that can be used to quickly find the sum of the interior angles of any polygon.

Materials

  • Graffiti mat sheet with a triangle in the middle and written instruction (one per group): “What do you know about angles? Talk to your group and write down what you know.”
  • Printed instructions for the initial task and extension activity (optional)
    • “What is the sum of the interior angles of your hexagon? What strategy did you use to figure this out?”
    • “What pattern can you find that would help you find the sum of the interior angles quickly for any polygon?”
  • Download: Appendix A – different hexagons (pair of matching hexagons cut out from card stock paper for each group)
  • Download: Appendix B- all hexagons on one page (one copy for each student) 
  • Download: Appendix C – heptagon and decagon (cut out from card stock paper; 2 for each group) 
  • Download: Appendix D – a table organizing the relationship between the number of sides, the number of triangles in a polygon, and the sum of interior angles
  • Scissors; rulers; tape; glue; markers; and protractors

Lesson

Part 1: Introduction

  • divide students into groups of 4. Hand out graffiti mat sheet to each group. “We will be thinking about angles today and working with polygons. First, think about what you know about angles. Talk to your group and write down what you know about angles on your graffiti mat.”
  • This gives students the opportunity to activate prior knowledge and primes them to think about angles in the context of polygons: a closed figure with at least 3 straight sides and angles.
  • At this point, some students will remember that a triangle has three angles that add up to 180 degrees. This fact is necessary to solve the next challenge. Provide additional support (described alter) to groups that do not have that understanding.

Part 2: Lesson

  • Present the next challenge, using visuals to support the instruction. “Now that we have had a chance to refresh our minds about angles, let’s talk about polygons. A polygon is a closed figure with at least 3 straight sides and 3 angles. The triangle on your paper is a polygon. Now you will get a hexagon; a hexagon is a polygon with six sides. Each group will get a different hexagon. With your group, work on finding a strategy that lets you figure out the sum of the interior angles of your hexagon, without a protractor. The interior angles are the angles that are inside the closed figure.”
  • Give each group two congruent hexagons that are unique from other groups (Appendix A). Having two congruent hexagons will allow for easy handling and sharing. Also, since the graffiti paper is still out on the table, students can choose to trace their group’s hexagon so that they each have the image directly in front of them. 
  • Write the instruction on the board (or alternatively, print them out and give a copy to each group): “What is the sum of the interior angles of your hexagon? What strategy did you use to figure this out?”
  • Based on the knowledge that a triangle’s interior angles add up to 180 degree, group can find the total degrees of the interior angles by splitting the hexagon into triangles. There is a number of ways to do this and different hexagons will prompt different problem solving strategies. If students already know the sum of interior angles of a quadrilateral (or can deduce it), then this might be another strategy they use.
  • After students are given sufficient time for problem solving, ask them if they are ready to share. Give each group a copy of the paper that shows all the different hexagons the class has been working on (Appendix B). This visual will help students while they listen to their peers talk about “their” hexagon. Sharing can be as simple as stating if they have found the sum of the interior angles for their hexagon, sharing their strategy for solving, or discussing a problem they encountered.
  • The sharing time can help groups to see that all of their hexagons have internal angles that add to 720 degrees. Groups can continue to work with the hexagons if they want more time and practice to solidify their understanding. Groups who are ready can try the extension.

Strategies for assisting students:

  • If it is clear that a group is lacking the necessary information that a triangle’s angles add up to 180 degrees, help them develop this understanding. Scaffold this understanding by giving groups a few different kinds of construction paper triangles. Demonstrate that if you rip off the corners of the triangle and place them next to each other with adjacent sides touching and points meeting at one point, the exposed edges will create a straight line.
  • If students seem stumped, ask them, “How could you use your knowledge of the sum of the angles in a triangle to help you solve this?”
  • If students are having trouble using the shapes they’ve drawn inside their hexagons to come up with the sum of the interior angles of a hexagon, ask them, “Do all of your drawn polygons’ internal angles also make up the hexagon’s interior angles?” “Is there a way you could make that so?”
  • You can suggest that students make the interior angles of their hexagon with a colour to help them focus on those angles while they work on coming up with a strategy.
  • A group that finishes quickly can be asked, “Can the method you used work for other hexagons?” They can be given a page with other groups’ hexagons traced on it.
  • if a group finishes really quickly, they could move on to the extension activity.

Possible student strategies:

  • Divide the hexagon into 6 equilateral triangles
  • Divide the hexagon into squares and triangles
  • Divide the hexagon into irregular triangles that converge onto one vertex

Part 3: Extension

  • Give each group 2 heptagons, and 2 decagons (Appendix C). “Now that you have some ideas about how to find the sum of interior angles of a hexagon, extend your strategy to a few other polygons. Take a few minutes to work with your heptagons (7 sides) and decagons (10 sides) and see if there is a pattern that can help you find the sum of interior angles quickly for any polygon. You don’t need to use a protractor to solve this but if you’d like to use one to help or to check your answers, they are at the front.” 
  • Write the instruction on the board (or alternatively, print them out and give a copy to each group): “What pattern can you find that would help you find the sum of interior angles quickly for any polygon?” 
  • After sufficient time for investigation, bring the groups back together to share again. At this time, not all groups may have arrived at the same understanding. Allowing students to hear other groups’ thinking will be important for consolidating the concepts explored in the lesson.
  • Students can also benefit from organizing the information in tables that can facilitate students’ ability to notice patterns (Appendix D). 

Strategies for assisting students:

  • Groups that are struggling could be asked, “Do you notice any pattern between the number of sides in your polygon and the sum of its interior angles?” “How could that pattern help you find the sum of any polygon’s interior angles quickly?” They could even be asked, “Do you notice any pattern between the number of triangles you’ve made inside your polygon and the sum of its interior angles?”
  • students could also be given other polygons (quadrilaterals, pentagons, octagons, and nonagons to ‘complete their set’ of polygons from a triangle to a decagon.

Possible Extensions

  • Ask students to use their pattern to write an algorithm that can be applied to any polygon to determine the sum of interior angles. 
  • Ask students to disprove their algorithm by looking for a polygon that does not fit the mold. Give this group a written definition for a polygon. Our lesson only dealt with convex polygons and not concave polygons. This extension will show that the algorithm only applies to convex polygons.
  • Ask students if there is an algorithm associated with convex polygons.