Polygons
Keystone Geometry
1
Polygons
Definition: A closed figure formed by a finite number of coplanar
segments so that each segment intersects exactly two
others, but only at their endpoints.
These figures are not polygons
These figures are polygons
2
Classifications of a Polygon
Convex: No line containing a side of the polygon contains a point
in its interior
Concave:
A polygon for which there is a line
containing a side of the polygon and
a point in the interior of the polygon.
3
Classifications of a
Polygon
Regular:
Irregular:
A convex polygon in which all interior angles have the
same measure and all sides are the same length
Two sides (or two interior angles) are not congruent.
4
Polygon Names
Classified by the number of sides
3 sides
4 sides
5 sides
6 sides
7 sides
8 sides
9 sides
10 sides
12 sides
n sides
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Dodecagon
n-gon
5
Convex Polygon Formulas…..
Diagonals of a Polygon: A segment connecting nonconsecutive
vertices of a polygon
For a convex polygon with n sides:
•
•
•
•
The more sides a polygon has, the more diagonals it will have.
A square has 2 diagonals.
A pentagon has 5 diagonals.
A hexagon has 9 diagonals………
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Examples
Diagonals in a square, n = 4
Diagonals in an octagon, n = 8
** What about a triangle?
A triangle does not have ANY
diagonals.
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Polygon Angle Formulas
Finding the INTERIOR angles in a convex polygon with n
sides:
The sum of the interior angles is (n - 2 )×180
The measure of one interior angle is (n - 2 )×180
n
Finding the EXTERIOR angles in a convex polygon with n
sides:
The sum of the exterior angles is 360°
The measure of one exterior angle is 360
n
8
Solving for the Angles
Polygon
Triangle
Quadrilateral
Pentagon
Hexagon
Interior Angle
Sum
180
360
540
720
Each Interior
Angle
60
90
108
120
Exterior
Angle Sum
360
360
360
360
Each Exterior
Angle
120
90
72
60
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Solving for the Angles
Polygon
Heptagon
Octagon
Nonagon
Decagon
Interior Angle
Sum
Each Interior
Angle
Exterior
Angle Sum
Each Exterior
Angle
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Examples…..
1. Sum of the measures of the interior angles of a 11-gon is
(n – 2)180°  (11 – 2)180 °  1620
2. The measure of an exterior angle of a regular octagon is
360 360
=
= 45
n
8
3. The number of sides of regular polygon with exterior angle 72 ° is
360
360
n=
®n=
=5
exterior angle
72
4. The measure of an interior angle of a regular polygon with 30 sides
( n-2) i180 = (30-2)i180 = 28i180 = 168
n
30
30
11