1

PolygonsKeystone Geometry

2

These figures are not polygons These figures are 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.

Polygons

3

Classifications of a PolygonConvex: 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

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.

Classifications of a Polygon

5

Polygon NamesClassified by the number of sides

3 sides Triangle4 sides5 sides6 sides7 sides8 sides

NonagonOctagonHeptagonHexagonPentagonQuadrilateral

10 sides9 sides

12 sides

DecagonDodecagon

n sides n-gon

6

Convex Polygon Formulas…..Diagonals of a Polygon:

For a convex polygon with n sides:

A segment connecting nonconsecutive vertices of a polygon

• 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………

Examples

7

Diagonals in a square, n = 4

Diagonals in an octagon, n = 8

** What about a triangle?A triangle does not have ANY diagonals.

8

Polygon Angle FormulasFinding the INTERIOR angles in a convex polygon with n sides:

The sum of the interior angles is

The measure of one interior angle is

The sum of the exterior angles is

The measure of one exterior angle is

Finding the EXTERIOR angles in a convex polygon with n sides:

Solving for the Angles

9

Polygon Triangle Quadrilateral Pentagon Hexagon

Interior Angle Sum

180 360 540 720

Each Interior Angle

60 90 108 120

Exterior Angle Sum

360 360 360 360

Each Exterior Angle

120 90 72 60

Solving for the Angles

10

Polygon Heptagon Octagon Nonagon Decagon

Interior Angle Sum

Each Interior Angle

Exterior Angle Sum

Each Exterior Angle

11

Examples…..1. Sum of the measures of the interior angles of a 11-gon is

(n – 2)180° (11 – 2)180 ° 1620

2. The measure of an exterior angle of a regular octagon is

3. The number of sides of regular polygon with exterior angle 72 ° is

4. The measure of an interior angle of a regular polygon with 30 sides