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Students will:

Students will: . Create a formula to find the interior angle sum given the number of sides . Adding It All Up. Polygon: a closed plane figure made of line segments Convex: the measures of all interior angles are less than 180°

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Students will:

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  1. Students will: Create a formula to find the interior angle sum given the number of sides

  2. Adding It All Up • Polygon: a closed plane figure made of line segments • Convex: the measures of all interior angles are less than 180° • Concave: the measure of at least one interior angle is greater than 180° • Regular: all angles and sides are congruent • Triangulation: the process of drawing diagonals (segments between non-adjacent vertices) to divide a polygon into non-overlapping triangles

  3. Draw these pentagons:

  4. Use the applet to examine pentagons, hexagons, heptagons and octagons, too. Do you notice a pattern? How does the sum of the angles change as the number of sides changes?

  5. As the number of sides increases, what happens to the sum of the angle measures? • For each additional side in a polygon, 180° is added to the sum of the angle measures.

  6. All polygons can be broken up into non-overlapping triangles

  7. Fill in the table as you work through each part of the activity.

  8. When you begin with a polygon with four or more sides and draw all the diagonals possible from one vertex, the polygon then is divided into several nonoverlapping triangles.

  9. The interior angle sum of this polygon can now be found by multiplying the number of triangles by 180°.

  10. If a convex polygon has n sides, then its interior angle sum is given by the following equation: S = ( n −2) × 180°.

  11. Find the sum of the measures of the interior angles of an octagon.

  12. Solution: • An octagon has 8 sides. So, n = 8. • Substitute 8 for n in the formula. • The sum of the measures of the interior angles of an octagon = (8 - 2)180°. • = 6 ×180° • = 1080° • The sum of the measures of the interior angles of an octagon is 1080°.

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