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

polygon A closed plane figure formed by three or more noncollinear straight lines that intersect only at their endpoints.

polygons

notpolygons

vertex

diagonal

regular

The common endpoint of two sides. Plural: verticesverticesverticesvertices.

A segment that connects any two nonconsecutive vertices.

A polygon that is both equilateral and equiangular.

vertexdiagonal

Polygons are named by the number of their sides:

SidesSidesSidesSides NameNameNameName TrianglesTrianglesTrianglesTriangles Sum Int.Sum Int.Sum Int.Sum Int.Each Int.Each Int.Each Int.Each Int.(Regular)(Regular)(Regular)(Regular)

3333 Triangle

4444 Quadrilateral

5555 Pentagon

6666 Hexagon

7777 Heptagon

8888 Octagon

9999 Nonagon

10101010 Decagon

12121212 Dodecagon

nnnn n-gon

Examples Identify the general name of each polygon:

1.

2.

3.

Examples Identify the general name of each polygon:

1.

2.

3.

pentagon

dodecagon

quadrilateral

concave

convex

A diagonal of the polygon contains points outside the polygon. (caved in)

Not concave.

concave pentagon

convex quadrilateral

outside the polygon

We know that the angles of a triangle add up to 180, but what about other polygons?

Draw a convex polygon of at least 4 sides:

We know that the angles of a triangle add up to 180, but what about other polygons?

Draw a convex polygon of at least 4 sides:

Now, draw all possible diagonals from one vertex. How many triangles are there?

What is the sum of their angles?

180180180180

180180180180

180180180180

Thm 6-1-1 Polygon Angle Sum Theorem

The sum of the interior angles of a convex polygon with n sides is

(n 2)180

If the polygon is equiangular, then the measure of one angle is

( 2)180n

n

Lets fill out the rest of the table.

SidesSidesSidesSides NameNameNameName TrianglesTrianglesTrianglesTriangles Sum Int.Sum Int.Sum Int.Sum Int.Each Int.Each Int.Each Int.Each Int.(Regular)(Regular)(Regular)(Regular)

3333 Triangle 1 (1)180=180 60

4444 Quadrilateral 2 (2)180=360 90

5555 Pentagon 3 (3)180=540 108

6666 Hexagon

7777 Heptagon

8888 Octagon

9999 Nonagon

10101010 Decagon

12121212 Dodecagon

nnnn n-gon

SidesSidesSidesSides NameNameNameName TrianglesTrianglesTrianglesTriangles Sum Int.Sum Int.Sum Int.Sum Int.Each Int.Each Int.Each Int.Each Int.(Regular)(Regular)(Regular)(Regular)

3333 Triangle 1 (1)180=180 60

4444 Quadrilateral 2 (2)180=360 90

5555 Pentagon 3 (3)180=540 108

6666 Hexagon 4 (4)180=720 120

7777 Heptagon 5 (5)180=900 128.6

8888 Octagon 6 (6)180=1080 135

9999 Nonagon 7 (7)180=1260 140

10101010 Decagon 8 (8)180=1440 144

12121212 Dodecagon 10 (10)180=1800 150

nnnn n-gon n 2 (n 2)180 ( 2)180n

n

An exterior angle is an angle created by extending the side of a polygon:

Now, consider the exterior angles of a regular pentagon:

Exterior angle

From our table, we know that each interior angles is 108. This means that each exterior angle is 180 108 = 72.

The sum of the exterior angles is therefore 5(72) = 360. It turns out this is true for any convex polygon, regular or not.

108

72

72

7272

72

Polygon Exterior Angle Sum Theorem

The sum of the exterior angles of a convex polygon is 360.

For any equiangular convex polygon with n sides, each exterior angle is

360

n

SidesSidesSidesSides NameNameNameName Sum Ext.Sum Ext.Sum Ext.Sum Ext. Each Ext.Each Ext.Each Ext.Each Ext.

3333 Triangle 360 120

4444 Quadrilateral 360 90

5555 Pentagon 360 72

6666 Hexagon 360 60

8888 Octagon 360 45

nnnn n-gon 360 360/n