Math 1312
Section 2.5
Convex Polygons
Definitions:
A polygon is a closed plane figure bounded by straight line segments as sides.
A convex polygon has two properties:
a) Every angle measures between 0o and 180o .
b) A line segment joining two points of a convex polygon remains inside or on the
boundary of the polygon.
A concave polygon:
a) Always has at least one reflex (between 180o and 360o ) angle.
b) A line segment joining two points of a concave polygon can contain points in the
exterior of the polygon.
Example 1:
Convex polygon
Concave Polygon
Special names for polygons with fixed numbers of sides
Number of sides
3
4
5
6
7
8
9
10
21
Polygon
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
21-gon
Not a polygon
Definition:
A diagonal of a polygon is a line segment that joins two nonconsecutive vertices.
Example 2:
B
A
C
D
Quadrilateral
4 sides
2 diagonals
Theorem: The total number of diagonals D in a polygon of n sides is given by the
n(n − 3)
formula D =
.
2
Example 3: Find the number of diagonals for any decagon.
Theorem: The sum of measures of the interior angles of a polygon with n sides is given
by S = (n − 2) ⋅ 180o .
Example 4: Find the sum of measures of the interior angles of a pentagon.
Example 5: Find the number of sides in a polygon whose sum of interior angles is 1980o .
Definition:
A regular polygon is a polygon that is both equilateral (all sides are congruent) and
equiangular (all angles are congruent).
Corollary: The measure I of each interior angle of a regular polygon of n sides is
(n − 2) ⋅ 180o
I=
.
n
Example 6: Find the measure of each interior angle of a stop sign.
Example 7: Find the number of sides that a regular polygon has if the measure of each
interior angle is 144o .
Corollary: The sum of the four interior angles of a quadrilateral is 360o .
Example 8: The angle sum of a square is 360o .
Corollary: The measure E of each exterior angle of a regular polygon of n sides is
360o
E=
.
n
Example 9: Find the measure of each exterior angle of a regular nonagon.
Example 10: Name the regular polygon whose exterior angle measures 120o .
Observation: An interior of a polygon angle and an adjacent exterior angle are
supplementary.
1
2
Example 11: If an interior angle of a regular polygon measures 165o , find
a) the measure of an exterior angle
b) the number of sides