Obj. 25 Properties of Polygons
- 1. Obj. 25 Properties of Polygons The student is able to (I can): • Name polygons based on their number of sides • Classify polygons based on — concave or convex — equilateral, equiangular, regular • Calculate and use the measures of interior and exterior angles of polygons
- 2. polygon A closed plane figure formed by three or more noncollinear straight lines that intersect only at their endpoints. polygons not polygons
- 3. vertex The common endpoint of two sides. Plural: vertices vertices. diagonal A segment that connects any two nonconsecutive vertices. diagonal regular vertex A polygon that is both equilateral and equiangular.
- 4. Polygons are named by the number of their sides: Sides Name 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon 12 Dodecagon n n-gon
- 5. Examples Identify the general name of each polygon: 1. pentagon 2. dodecagon 3. quadrilateral
- 6. concave A diagonal of the polygon contains points outside the polygon. (“caved in”) convex Not concave. concave pentagon convex quadrilateral
- 7. We know that the angles of a triangle add up to 180º, but what about other polygons? Draw a convex polygon of at least 4 sides: 180º 180º 180º Now, draw all possible diagonals from one vertex. How many triangles are there? What is the sum of their angles?
- 8. Thm 6-1-1 Polygon Angle Sum Theorem The sum of the interior angles of a convex polygon with n sides is (n — 2)180º If the polygon is equiangular, then the measure of one angle is (n − 2)180° n
- 10. Let’s update our table: Sides Name Triangles Sum Int. Each Int. (Regular) 3 Triangle 1 (1)180º=180º 60º 4 Quadrilateral 2 (2)180º=360º 90º 5 Pentagon 3 (3)180º=540º 108º 6 Hexagon 4 (4)180º=720º 120º 7 Heptagon 5 (5)180º=900º ≈128.6º 8 Octagon 6 (6)180º=1080º 135º 9 Nonagon 7 (7)180º=1260º 140º 10 Decagon 8 (8)180º=1440º 144º 12 Dodecagon 10 (10)180º=1800º 150º n n-gon n—2 (n — 2)180º (n − 2)180° n
- 11. An exterior angle is an angle created by extending the side of a polygon: Exterior angle Now, consider the exterior angles of a regular pentagon:
- 12. From our table, we know that each interior angles is 108º. This means that each exterior angle is 180 — 108 = 72º. 72º 72º 72º 108º 72º 72º The sum of the exterior angles is therefore 5(72) = 360º. It turns out this is true for any convex polygon, regular or not.
- 13. Polygon Exterior Angle Sum Theorem The sum of the exterior angles of a convex polygon is 360º. For any equiangular convex polygon with n sides, each exterior angle is 360° n Sides Name Sum Ext. Each Ext. 3 Triangle 360º 120º 4 Quadrilateral 360º 90º 5 Pentagon 360º 72º 6 Hexagon 360º 60º 8 Octagon 360º 45º n n-gon 360º 360º/n