Polygons
A many sided figure
The cross section of a
brilliant-cut diamond
forms a pentagon. The
most beautiful and
valuable diamonds have
precisely cut angles
that maximize the
amount of light they
reflect.
A pentagon is a type of
polygon.
Prefixes are used to
name different types
of polygons.
Polygon – a closed plane figure formed by
three or more segments.
Regular polygon – a polygon with congruent
sides and angles.
Prefixes used to name polygons: tri-, quad-,
penta-, hexa-, hepta-, octa-, nona-, decaPolygons are named (classified) based on the
number of sides.
Polygons
Properties of polygons, interior angles of
polygons including triangles, quadrilaterals,
pentagons, heptagons, octagons, nonagons, and
decagons.
Properties of Triangles
Triangle – 3-sided polygon
The sum of the angles in any triangle is
180° (triangle sum theorem)
The formula we use to find the sum of the
interior angles of any polygon comes from the
number of triangles in a figure
First remember that the
sum of the interior angles of
a polygon is given by the
formula 180(n-2).
A polygon is called a
REGULAR when all the sides
are congruent and all the
angles are congruent.
The picture shown to the
left is that of a Regular
Pentagon. We know that to
find the sum of its interior
angles we substitute n = 5 in
the formula and get:
180(5 -2) = 180(3) = 540°
Regular triangles - Equilateral
All sides are the same length
(congruent) and all interior
angles are the same size
(congruent).
To find the measure of the
interior angles, we know that
the sum of all the angles equal
180°, and there are three
angles.
So, the measure of the interior
angles of an equilateral triangle
is 60°.
Quadrilaterals – squares
All sides are the same length
(congruent) and all interior
angles are the same size
(congruent)
To find the measure of the
interior angles, we know that the
sum of the angles equal 360°,
and there are four angles, so the
measure of the interior angles
are 90°.
Pentagon – a 5-sided
polygon
To find the sum of the
interior angles of a
pentagon, we divide the
pentagon into triangles.
There are three triangles
and because the sum of
each triangle is 180° we
get 540°, so the measure
of the interior angles of a
regular pentagon is 540°
Hexagon – a 6-sided
polygon
To find the sum of the
interior angles of a hexagon
we divide the hexagon into
triangles. There are four
triangles and because the
sum of the angles in a
triangle is 180°, we get
720°, so the measure of the
interior angles of a regular
hexagon is 720°.
Octagon – an 8-sided polygon
All sides are the same length (congruent) and
all interior angles are the same size
(congruent).
What is the sum of the angles in a regular
octagon?
Nonagon – a 9-sided
polygon
All sides are the same
length (congruent) and
all interior angles are
the same size
(congruent).
What is the sum of the
interior angles of a
regular nonagon?
Decagon – a 10-sided
polygon
All sides are the same
length (congruent) and all
interior angles are the
same size (congruent).
What is the sum of the
interior angles of a regular
decagon?
Using the pentagon example, we can
come up with a formula that works
for all polygons.
Notice that a pentagon has 5
sides, and that you can form 3
triangles by connecting the
vertices. That’s 2 less than the
number of sides. If we represent
the number of sides of a polygon as
n, then the number of triangles you
can form is (n-2). Since each
triangle contains 180°, that gives
us the formula:
sum of interior angles = 180(n-2)
Warning !
• Look at the pentagon to the
right. Do angle E and angle
B look like they have the
same measures? You’re
right---they don’t. This
pentagon is not a regular
pentagon.
•
If the angles of a polygon do not all
have the same measure, then we
can’t find the measure of any one of
the angles just by knowing their sum.
Using the Formula
Example 1: Find the number of degrees in the sum
of the interior angles of an octagon.
An octagon has 8 sides. So n = 8. Using our formula,
that gives us 180(8-2) = 180(6) = 1080°
Example 2: How many sides does a polygon
have if the sum of its interior angles is 720°?
Since, this time, we know the number of
degrees, we set the formula equal to 720°,
and solve for n.
180(n-2) = 720 set the formula = 720°
n - 2 = 4 divide both sides by 180
n=6
add 2 to both sides
Names of Polygons
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon or Septagon
Octagon
Nonagon or Novagon
Decagon
3 sides
4 sides
5 sides
6 sides
7 sides
8 sides
9 sides
10 sides
Practice with Sum of Interior
Angles
1)
The sum of the interior angles of a hexagon.
a)
b)
c)
360°
540°
720°
2)
How many degrees are there in the
sum of the interior angles of a 9-sided
polygon?
a) 1080°
b) 1260°
c) 1620°
3)
If the sum of the interior angles of a
polygon equals 900°, how many sides
does the polygon have?
a) 7
b) 9
c) 10
4)
How many sides does a polygon have if
the sum of its interior angles is 2160°?
a) 14
b) 16
c) 18
5)
What is the name of a polygon if the sum
of its interior angles equals 1440°?
a) octagon
b) decagon
c) pentagon
Special Quadrilaterals
4-sided figures
Quadrilaterals with certain
properties are given additional
names.
A square has 4
congruent sides and 4
right angles.
A rectangle has 4 right angles.
A parallelogram has 2
pairs of parallel sides.
A rhombus has
4 congruent
sides.
A kite has 2 sets
of adjacent sides
that are the same
length (congruent)
and one set of
opposite angles
that are
congruent.
Algebra in Geometry
Applying Geometric Properties
Algebra can be used to solve many problems in
geometry. Using variables and algebraic
expressions to represent unknown measures
makes solving many problems easier.
Find the sum of interior angles
using the formula.
180°(n - 2) = 180°(4 – 2) =
180°(2) = 360°
Set the sum of the angles equal
to the total.
120° + 50° + 80° + x = 360°
250° + x = 360°
250 – 250 + x = 360 -250
x = 110°
Remember, a regular polygon
has congruent sides and
congruent angles.
Given the regular pentagon at
the left, what are the
measures of the interior
angles. (use the formula)
180°(n – 2) =
180°(5 – 2) =
180°(3) = 540°
# of angles = 5
540°/5 = 108°
Each angle in a regular
pentagon measures 108°
Using geometry to solve word problems.
Remember, draw a picture.
Quadrilateral STUV has angle
measures of:
(3x + 15)°
(2x + 20)°
(4x + 5)°
(2x – 10)°, add the angles = 360
(3x + 15) + (2x + 20) + (4x + 5)
+ (2x – 10) = 360
11x + 30 = 360
11x = 330
x = 30°
x = 30°, then
3x + 15 = 3(30) + 15 = 105°
2x + 20 = 2(30) + 20 = 80°
4x + 5 = 4(30) + 5 = 125°
2x – 10 = 2(30) – 10 = 50°
So,
105° + 80° + 125° + 50° = 360°
Solve the following:
Figure ABCDEF is a convex polygon with the
following angle measures. What is the measure of
each angle? (draw a picture)
A = 4x
B = 2x
C = 3x
D = 5x + 10
E = 3x – 20
F = 2x – 30
Answer »»
(4x) + (2x) + (3x) + (5x + 10) + (3x – 20) + (2x – 30) = 720°
19x – 40 = 720°
19x = 720°
x = 40°, so
4x = 4(40) = 160°
2x = 2(40) = 80°
3x = 3(40) = 120°
5x + 10 = 5(40) + 10 = 210°
3x – 20 = 3(40) – 20 = 100°
2x – 30 = 2(40) – 30 = 50°
check,
160° + 80° + 120° + 210° + 100° + 50° = 720°
720° = 720°
Polygons
Problem Solving
1) Find the sum of the
angle measures in the
figure to the left.
a)
b)
c)
d)
180°
540°
720°
1260°
2) Find the angle
measures in the
polygon to the right.
a)
b)
c)
d)
m° = 150°
m° = 144°
m° = 120°
m° = 90°
3) Give all the names that apply
to the figure at the left.
a) quadrilateral, square,
rectangle, rhombus,
parallelogram
b) quadrilateral, trapezoid
c) quadrilateral, parallelogram,
rectangle, square
d) quadrilateral, parallelogram,
trapezoid
4) Find the sum of the angle measures in a 20-gon.
If the polygon is regular, find the measure of each
angle.
a)
b)
c)
d)
198°, 9.9°
720°, 72°
1800°, 90°
3240°, 162°
5) Find the value of
the variable.
a)
b)
c)
d)
x° = 90°
x° = 110°
x° = 120°
x° = 290°
6) Given the
polygon at the left,
what is the measure
of the interior
angles?
A)
B)
C)
D)
720
540
360
180