7.3 FORMULAS INVOLVING POLYGONS

Objective:After studying this section, you

will be able to use some important formulas that apply to polygons.

A polygon with 3 sides can be called a 3-gon, a seven sided polygon can be called a 7-gon. Many have special names.

Number of Sides (or vertices)

Polygon

34567891

01215n

TriangleQuadrilateralPentagonHexagonHeptagonOctagonNonagonDecagonDodecagonPentadecagonN-gon

How do we find the sum of all the angles in a polygon with n sides?

To answer this question, let’s use what we know. We know that the sum of all the angles in a 3 sided polygon is 180.

Draw a 5-sided polygon in your notes.

Pick one vertex and draw all the diagonals from that vertex that you can.

How many triangles did you make?

Since we know that the sum of all the angles in a triangle equals 180 and there are three triangles in a pentagon, multiply 3(180) and you will have the sum of all the angles.

Try finding the sum of the angles in a heptagon.

Theorem

The sum Si of the measures of the angles of a polygon with n sides is given by the formula Si = 180(n – 2)From time to time we may refer to the angles of a polygon as the interior angles of the polygon.

A1

2

B

C3D

4

5E

In the diagram exterior angles have been formed by extending the side of the polygon at each vertex.

At vertex A, . We can add each exterior angle to its adjacent angle, getting a sum of 180 at each vertex. Since there are five vertices we can calculate the total sum as 5(180) = 900.

1 180m m EAB

With the sum of the interior angles in a pentagon being 3(180) or 540, if we subtract that from 900 we will have the sum of the exterior angles of a pentagon.

The sum of the measures of the exterior angles of a pentagon is 900 – 540 = 360.

Find the sum of the measures of the exterior angles in a hexagon.

If one exterior angle is taken at each vertex, the sum Se of the measures of the exterior angles of a polygon is given by the formula Se = 360.

Theorem

TheoremThe number d of diagonals that can be drawn in a polygon of n sides is given by the formula

( 3)

2

n nd

Example 1

Find the sum of the measures of the interior angles of the figure.

Example 2

Example 3

Find the number of diagonals that can be drawn in a pentadecagon.

What is the name of the polygon if the sum of the measures of the angles is 1080?

SummaryExplain in your own words what the formula Si = 180(n – 2) means. What specifically does (n – 2) represent?

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