1. Polygons Keystone Geometry

2. Polygons
Definition:
A closed figure formed by a finite number of coplanar segments so that each segment intersects exactly two others, but only at their endpoints.
These figures are not polygons
These figures are polygons

3. Classifications of a Polygon
Convex:
No line containing a side of the polygon contains a point in its interior
Concave:
A polygon for which there is a line containing a side of the polygon and a point in the interior of the polygon.

4. Classifications of a Polygon
Regular:
A convex polygon in which all interior angles have the same measure and all sides are the same length
Irregular:
Two sides (or two interior angles) are not congruent.

5. Polygon Names Classified by the number of sides
Triangle
4 sides
Quadrilateral
5 sides
Pentagon
6 sides
Hexagon
7 sides
Heptagon
8 sides
Octagon
9 sides
Nonagon
10 sides
Decagon
Dodecagon
12 sides
n sides
n-gon

6. Convex Polygon Formulas…..
Diagonals of a Polygon:
A segment connecting nonconsecutive vertices of a polygon
For a convex polygon with n sides:
The more sides a polygon has, the more diagonals it will have.
A square has 2 diagonals.
A pentagon has 5 diagonals.
A hexagon has 9 diagonals………

7. Examples Diagonals in a square, n = 4 Diagonals in an octagon, n = 8
** What about a triangle?
A triangle does not have ANY
diagonals.
Diagonals in an octagon, n = 8

8. Polygon Angle Formulas
Finding the INTERIOR angles in a convex polygon with n sides:
The sum of the interior angles is
The measure of one interior angle is
Finding the EXTERIOR angles in a convex polygon with n sides:
The sum of the exterior angles is
The measure of one exterior angle is

9. Solving for the Angles Polygon Triangle Quadrilateral Pentagon Hexagon
Interior Angle Sum
180
360
540
720
Each Interior Angle 60 90
108
120
Exterior Angle Sum
Each Exterior Angle 72

10. Solving for the Angles Polygon Heptagon Octagon Nonagon Decagon
Interior Angle Sum
Each Interior Angle
Exterior Angle Sum
Each Exterior Angle

11. Examples….. Sum of the measures of the interior angles of a 11-gon is
The measure of an exterior angle of a regular octagon is
The number of sides of regular polygon with exterior angle 72 ° is
The measure of an interior angle of a regular polygon with 30 sides