6th Grade Math Textbook, Fundamentals

convex polygons concave polygons

 &KDSWHU

9-7

Polygons

Objective To find the exterior and interior angles of polygons• To identify regular, convex, and

concave polygons• To find the sum of angle measures of polygons • To find the missing measure of

angles of a polygon

For an outdoor art project Sid is working on, he mows a large

field to create several large regular triangles. What are the

measures of each of the interior angles of his grass figures?

To find the measures, use the properties of polygons. A
is a closed plane figure that has three or more sides. Each side is
a line segment. The line segments meet only at their endpoints.
These are the vertices of the polygon.

Polygons have both interior and exterior angles.

lie inside a polygon. The number of

interior angle is the same as the number of vertices.

The sides of a polygon can be extended to form

that lie outside the polygon. Each

exterior angle is adjacent and supplementary to

an interior angle.

Polygons also can be either convex or concave. Each interior angle of a

measures less than 180. A contains one

or more interior angles that each have a measure greater than 180.

convex polygon concave polygon

exterior angles

Interior angles

polygon

A regular polygonhas congruent sides and congruent angles.

180°  x°

x°

AC

B

ED

interior angles exterior angles

Key Concept

An n-gon, a polygon with n

sides, has nvertices and n

angles. Polygons are named

by the letters of their vertices

in consecutive order.

A is a polygon with three sides and three vertices.

The sum of the interior angle measures for any triangle is 180.

You can show that the sum is 180 by cutting out the

angles and arranging them along a straight line.

Because the measure of a straight angle is 180 ,

the sum of these interior angle measures is also 180.

So each of the interior angles of Sid’s grass figures has a

measure of 180

3, or 60.

triangle

regular quadrilateral regular pentagon regular hexagon

side

vertex

All regular polygons

are convex polygons.