Polygon Angles

Today, we first reviewed the names of polygons with 3, 4, 5, 6, 7, 8, 9, 10, or 12 sides (any other number of sides to a polygon (#) is just called by #-gon). Then we explored how to break up all of the shapes, no matter how many sides they have, to have only triangles inside them. Because we know that the sum of the measures of the three angles inside a triangle is 180°, we can figure out the sum of the measures of the angles inside any polygon based on how many triangles it takes to fill the same shape.

Example: A pentagon can be filled with 3 triangles:

Because each triangle has 180° total for its three angles, then the 3 triangles have 3 * 180° or 520° for the 5 angles inside the pentagon.

Students discovered that there is relationship between the number of triangles it takes and the number of sides: There are always 2 fewer triangles to fill the shape than there are sides. Using this relationship, students can find the sum of the interior angles of any convex polygon by using the formula 180°(n - 2), where n is the number of sides.

Example: Find the sum of the interior angles of a 23-gon.

A 23-gon has 23 sides, so 21 triangles are needed to fill the shape.
180° * 21 = 3780°

Students also learned that the sum of the exterior angles of a convex polygon is always 360° (no matter how many sides it has). The model for this that I gave students was to consider walking around the shaping and turning to follow the next corner. As I made my way around my desk, I had to turn all the way around by the time I got back to the beginning. They know that all the way around a full circle is 360°

Example: There are 4 exterior angles shown in this quadrilateral.

Homework: Page 300, #2 - 24 even; Page 302, #2 - 14 even, #34.

There will be a quiz this week. It may be as early as Thursday. It will cover the lessons on angles (both interior and exterior) given on Friday, today, and tomorrow. There will be a review session Wednesday morning in room 4303 for this quiz.

Remember: Vocabulary definitions are due on Wednesday!
Remember, also: The Logic Project is due on Friday!

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