Polygons 6-1

Definition of PolygonA polygon is a closed figure formed by an finite number of coplanar segments

such that the sides that have a common endpoint are non-collinear each side intersects exactly two other sides, but only at their

endpoints. Symbolic Representation: Is named by the letters of it’s vertices, in

consecutive order.

Polygons Not Polygons

A convex polygon is a polygon such that no line containing a side of the polygon contains a point in the interior of the polygon.

A polygon that is not convex is called concave.

ConvexConcave

Polygons may be classified by the number of sides they have. In general, a polygon with n sides is called an n-gon. This means the nonagon can also be called a 9-gon.

Number of Sides

Polygon

3 triangle4567891012n

Polygons may be classified by the number of sides they have. In general, a polygon with n sides is called an n-gon. This means the nonagon can also be called a 9-gon.

Number of Sides

Polygon

3 triangle4 Quadrilatera

l5 Pentagon6 Hexagon7 heptagon8 Octagon9 Nonagon10 Decagon12 Dodecagonn N-gon

Perimeter

Is the sum of the lengths of its sides.

Perimeter of Special shapes:

Triangle Square Rectangle

cbap sp

ssssp

4

lwp

llwwp

22

Regular polygon

a convex polygon with all angles and all sides congruent.

Find the Perimeter

Find the Perimeter using the coordinate plane:

P(-5, 1), Q(-1, 4), R(-6, -8)Step 1: Graph the points.Step 2: Use distance formula on all sides.Step 3: Add the three sides.

Diagonal of a polygon

a segment drawn from one vertex of a polygon to a nonconsecutive vertex.

Quadrilateral _________________ _________________ _________________

Notice in each case before this the polygon is separated into triangles. The sum of the measures of the angles of each polygon can be found by adding the measures of the angles of the triangles. This is easy to find since the sum of the angles in a triangle = ______.

Use the chart below to find a pattern:

Convex Polygon

# of sides # of triangles

Sum of measures

triangle 3 1 1(180) = 180

quadrilateral 4 2 2(180) = 360

pentagon

hexagon

heptagon

octagon

n-gon

Find the measure of each interior angle

Interior Angle Sum Theorem

If a convex polygon has n sides and S is the sum of the measures of its interior angles, then

S = 180(n – 2).

Exterior Angle Sum Theorem

If a polygon is convex, then the sum of the measures of the exterior angles, one at each vertex, is 360.

Consider the following regular polygons. Find the sum of the exterior angles

sum of int. angle=

Find the sum of the measures of the interior angles of each convex polygon:1. decagon 2. 21-gon

3. 13-gon 4. 46-gon

The measure of an exterior angle of a regular polygon is given. Find the number of sides of the polygon.5. 30 6. 8

7. 72 8. 14.4

The number of sides of a regular polygon is given. Find the measures of an interior angle and an exterior angle for each polygon. Round to the nearest hundredth9. 30 10. 9

11. 22 12. 14

The measure of an interior angle of a regular polygon is given. Find the number of sides in each polygon.13. 135 14. 144

15. 176.4 16. 165.6