Transcript Polygons
Section 3.5 Polygons WHAT IS A POLYGON? A polygon is: A closed plane figure made up of several line segments they are joined together. The sides to not cross each other. Exactly two lines meet at every vertex. Polygons: Not Polygons: PARTS OF A POLYGON Side - one of the line segments that make up the polygon. Vertex - point where two sides meet. Two or more of these points are called vertices. Diagonal - a line connecting two vertices that isn't a side. Interior Angle - Angle formed by two adjacent sides inside the polygon. Exterior Angle - Angle formed by two adjacent sides outside the polygon. TYPES OF POLYGONS Convex Polygons: Nonconvex Polygons: A polygon such that: o Every interior angle is less than 180° o Every line segment between two vertices does not go on the exterior of the polygon. (It remains inside or on the boundaries of the polygon) A polygon such that: o At least one interior angle has measure greater than 180° o There exists a line segment between two vertices is on the exterior of the polygon. TYPES OF POLYGONS Polygons are classified by the number of sides they have. Number of Sides Name 3 4 5 Triangle Quadrilateral Pentagon 6 7 8 10 Hexagon Heptagon Octagon Decagon Diagonals of a Polygon To find the number of diagonals in each polygon we need to know n, the number of sides Use the formula: n(n 3) 2 ANGLES OF A POLYGON Suppose you start with a pentagon. If you pick any vertex of that figure, and connect it to all the other vertices, how many triangles can you form? If you start with vertex A and connect it to all other vertices (it's already connected to B and E by sides) you form three triangles. Each triangle contains 1800. So the total number of degrees in the interior angles of a pentagon is: 3 x 180° = ANGLES OF A POLYGON We can apply this to any convex polygon. The sum of the measure of the angles of a convex polygon with n sides is: (n-2)180 We can also use this formula to find the number of sides in a polygon. EXAMPLES: Example 1: Find the number of degrees in the sum of the interior angles of an octagon. An octagon has 8 sides. So n = 8. Using the formula, that gives us (8-2)180= (6)180 = 1080° Example 2: Find the number of degrees in the sum of the interior angles of a quadrilateral. A quadrilateral has 4 sides. So n = 4. Using the formula, that gives us (4-2)180= (2)180 = 360° EXAMPLES: Example 3: How many sides does a polygon have if the sum of its interior angles is 7200 ? Since, this time, we know the number of degrees, we set the formula equal to 720°, and solve for n. (n-2) 180 = 720 n-2 = 4 n= 6 REGULAR POLYGONS A polygon that is both equiangular (all angles congruent) and equilateral (all sides congruent). Find the measure of each interior angle and exterior angle of a regular hexagon: The sum of interior angles is (6-2)180=720 Since all six angles are congruent, each interior angle has a measure of 720/6=120 120 ° 120 ° Since each polygon has exterior angles that add to 360, each exterior angle has measure 360/6=60 EXTERIOR ANGLE MEASURES The sum of the measures of the exterior angles of any convex polygon, one at each vertex, is 360° An exterior angle of a polygon is formed by extending one side of the polygon. See Geometer’s Sketchpad for Demonstration. EXAMPLES Find the sum of the exterior angles of : A Pentagon: A Decagon: 360° 360° A 15-Sided Polygon: A 7-Sided Polygon: Remember... 360° 360° The sum of the measures of the exterior angles of any convex polygon, one at each vertex, is always 360°! Example Using an Octagon # of Diagonals: n(n 2) 8(8 2) 8 6 48 16 3 Sum of the Interior Angles: Each Interior Angle: 3 3 3 (n 2)180 (8 2)180 6 180 1080 1080 135 8 Sum of the Exterior Angles: 360 Each Exterior Angle: 360 360 45 n 8 Example Using an Heptagon # of Diagonals: Sum of the Interior Angles: Each Interior Angle: Sum of the Exterior Angles: Each Exterior Angle: Example Using a Quadrilateral # of Diagonals: Sum of the Interior Angles: Each Interior Angle: Sum of the Exterior Angles: Each Exterior Angle: