Angles in a polygon Lecture 419

Transcription

1 Angles in a polygon Lecture 419

2 Formula For an n-sided polygon, the number of degrees for the sum of the internal angles is 180(n-2)º. For n=3 (triangle), it's 180. For n=4 (quadrilateral), it's 360. And so on. The formula we use to find the sum of the interior angles of any polygon comes from the following idea: Suppose you start with a pentagon. If you pick any vertex (the point where any 2 sides meet) 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 180º. So the total number of degrees in the interior angles of a pentagon is: 3 x 180º = 540º 2

3 Polygons Triangle Quadrilateral Pentagon Hexagon Heptagon or Septagon Octagon Nonagon or Novagon Decagon 3 sides 4 sides 5 sides 6 sides 7 sides 8 sides 9 sides 10 sides 3

4 Types of polygons Regular - all angles are equal and all sides are the same length. Regular polygons are both equiangular and equilateral. Equiangular - all angles are equal. Equilateral - all sides are the same length. Convex - a straight line drawn through a convex polygon crosses at most two sides. Every interior angle is less than

5 Concave - you can draw at least one straight line through a concave polygon that crosses more than two sides. At least one interior angle is more than 180. A complex polygon: 5

6 Regular polygon The inner angle of a regular polygon is (n 2)π/n radians or (n 2)180 /n. Example Find the value of an interior angle of a 12-sided regular polygon. A 12-sided polygon is equivalent to 10 triangles. So 10 x 180 = Now, the polygon is regular which means all the angles are the same. We have 12 angles, so each much be 1800/12 = 150. Example Into how many triangles can a heptagon be divided? n = 7. Number of triangles = n 2 = 5. 6

7 Exterior angles The exterior angles of a polygon always add up to 360. The figure below shows the definition of the exterior angles of a polygon. In the given case there are 6 sides (hexagon). The sum of the interior angles equals 180 (n 2). = 180 (6 2) = = 720. The more sides a polygon has, the larger the sum of its internal angles. The larger the number of sides, the more closely a regular convex polygon approximates a circle. 7

8 Example Is it possible for a regular polygon to have an interior angle measure of 169? ( n 2) n Simplify the above to yield n = Since n is not a natural number, there is no regular polygon with interior angle measure of

9 Example Is it possible for a regular polygon to have an interior angle measure of 175.5? ( n 2) n Simplify to yield n = 80, a natural number. An 80-gon has an interior angle measure of

10 Example If n = 50, what is the measure of each interior angle of a regular polygon? 180(50 2) is

11 Example If n = 100, what is the measure of each interior angle of a regular polygon? 180(100 2) is

12 Example As n gets very large, (n ), what happens to the measure of an interior angle of a regular polygon? 180( 2) Interior angle of a regular polygon = n 360 = 180. As n n n becomes very large 360/n becomes zero, so this means that interior angle is equal to

13 Example As the number of sides of the polygon increases, what happens to the measures of each exterior angle of a regular polygon? As the number of sides of the polygon increases, the measure of each exterior angle approaches zero. This results because the measure of the supplementary interior angle approaches

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