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Polygons and Area

§ 10.1 Naming Polygons

§ 10.2 Diagonals and Angle Measure

§ 10.3 Areas of Polygons

§ 10.4 Areas of Triangles and Trapezoids

§ 10.5 Areas of Regular Polygons

§ 10.6 Symmetry

§ 10.7 Tessellations
Naming Polygons
You will learn to name polygons according to the number of
sides and ______.
angles
_____
1) regular polygon
2) convex
3) concave
Naming Polygons
closed figure in a plane formed by segments, called sides.
A polygon is a _____________
sides or ______.
angles
A polygon is named by the number of its _____
tri means three.
A triangle is a polygon with three sides. The prefix ___
Naming Polygons
Prefixes are also used to name other polygons.
Prefix
Number of
Sides
Name of
Polygon
tri-
3
triangle
quadri-
4
quadrilateral
penta-
5
pentagon
hexa-
6
hexagon
hepta-
7
heptagon
octa-
8
octagon
nona-
9
nonagon
deca-
10
decagon
Naming Polygons
A vertex is the point
of intersection of
two sides.
Consecutive vertices are
the two endpoints of any
side.
Q
P
R
U
A segment whose
endpoints are
nonconsecutive
vertices is a
diagonal.
T
S
Sides that share a vertex
are called consecutive
sides.
Naming Polygons
sides congruent.
An equilateral polygon has all _____
angles congruent.
An equiangular polygon has all ______
equilateral and ___________.
equiangular
A regular polygon is both ___________
equilateral
but not
equiangular
equiangular
but not
equilateral
regular,
both equilateral
and equiangular
Investigation: As the number of sides of a series of regular polygons increases, what do you
notice about the shape of the polygons?
Naming Polygons
A polygon can also be classified as convex or concave.
If all of the diagonals
lie in the interior of
the figure, then the
convex
polygon is ______.
If any part of a diagonal lies
outside of the figure, then the
concave
polygon is _______.
Naming Polygons
Diagonals and Angle Measure
You will learn to find measures of interior and exterior angles
of polygons.
Nothing New!
Diagonals and Angle Measure
Make a table like the one below.
1) Draw a convex quadrilateral.
2) Choose one vertex and draw all
possible diagonals from that vertex.
3) How many triangles are formed?
Convex
Polygon
Number
of Sides
quadrilateral
4
Number of Diagonals
from One Vertex
1
Number of
Triangles
2
Sum of
Interior Angles
2(180) = 360
Diagonals and Angle Measure
1) Draw a convex pentagon.
2) Choose one vertex and draw all
possible diagonals from that vertex.
3) How many triangles are formed?
Convex
Polygon
Number
of Sides
Number of Diagonals
from One Vertex
Number of
Triangles
Sum of
Interior Angles
quadrilateral
4
1
2
2(180) = 360
pentagon
5
2
3
3(180) = 540
Diagonals and Angle Measure
1) Draw a convex hexagon.
2) Choose one vertex and draw all
possible diagonals from that vertex.
3) How many triangles are formed?
Convex
Polygon
Number
of Sides
Number of Diagonals
from One Vertex
Number of
Triangles
Sum of
Interior Angles
quadrilateral
4
1
2
2(180) = 360
pentagon
5
2
3
3(180) = 540
hexagon
6
3
4
4(180) = 720
Diagonals and Angle Measure
1) Draw a convex heptagon.
2) Choose one vertex and draw all
possible diagonals from that vertex.
3) How many triangles are formed?
Convex
Polygon
Number
of Sides
Number of Diagonals
from One Vertex
Number of
Triangles
Sum of
Interior Angles
quadrilateral
4
1
2
2(180) = 360
pentagon
5
2
3
3(180) = 540
hexagon
6
3
4
4(180) = 720
heptagon
7
4
5
5(180) = 900
Diagonals and Angle Measure
1) Any convex polygon.
2) All possible diagonals from one vertex.
3) How many triangles?
Convex
Polygon
Number
of Sides
Number of Diagonals
from One Vertex
Number of
Triangles
Sum of
Interior Angles
quadrilateral
4
1
2
2(180) = 360
pentagon
5
2
3
3(180) = 540
hexagon
6
3
4
4(180) = 720
heptagon
7
4
5
5(180) = 900
n-gon
n
n-3
n-2
(n – 2)180
If a convex polygon has n sides, then the sum of the
Theorem 10-1 measure of its interior angles is (n – 2)180.
Diagonals and Angle Measure
In §7.2 we identified exterior angles of triangles.
Likewise, you can extend the sides of any
convex polygon to form exterior angles.
57°
The figure suggests a method for finding the
sum of the measures of the exterior angles
of a convex polygon.
48°
72°
54°
When you extend n sides of a polygon,
n linear pairs of angles are formed.
The sum of the angle measures in each linear pair is 180.
sum of measure of
exterior angles
sum of measure of
exterior angles
=
sum of measures of
linear pairs
–
sum of measures of
interior angles
=
=
n•180
180n
–
–
180(n – 2)
180n + 360
=
360
74°
55°
Diagonals and Angle Measure
In any convex polygon, the sum of the measures of the
Theorem 10-2
exterior angles, (one at each vertex), is 360.
Java Applet
Diagonals and Angle Measure
Areas of Polygons
You will learn to calculate and estimate the areas of polygons.
1) polygonal region
2) composite figure
3) irregular figure
Areas of Polygons
polygonal region
Any polygon and its interior are called a ______________.
In lesson 1-6, you found the areas of rectangles.
For any polygon and a given unit of measure, there is a
Postulate 10-1
unique number A called the measure of the area of the
Area Postulate
polygon.
Area can be used to describe, compare, and contrast polygons. The two
polygons below are congruent. How do the areas of these polygons compare?
They are the same.
Postulate 10-2
Congruent polygons have equal areas.
Areas of Polygons
composite figures
The figures above are examples of ________________.
They are each made from a rectangle and a triangle that have been placed
together. You can use what you know about the pieces to gain information
about the figure made from them.
You can find the area of any polygon by dividing the original region into
rectangles
squares __________,
smaller and simpler polygon regions, like _______,
triangles
and ________.
adding the
The area of the original polygonal region can then be found by __________
areas of the smaller polygons
_________________________.
Areas of Polygons
The area of a given polygon equals the sum of the areas of
the non-overlapping polygons that form the given polygon.
Postulate 10-3
Area Addition
Postulate
1
2
3
AreaTotal = A1 + A2 + A3
Areas of Polygons
Find the area of the polygon in square units.
Area of polygon =
Area of Square
3u X 3u = 9u2
Area of Rectangle
1u X 2u = 2u2
= 7u2
3 units
3 units
Area of
Area of Square
Rectangle
Areas of Polygons
Areas of Triangles and Trapezoids
You will learn to find the areas of triangles and trapezoids.
Nothing new!
Areas of Triangles and Trapezoids
Look at the rectangle below. Its area is bh square units.
congruent triangles
The diagonal divides the rectangle into two _________________.
1
The area of each triangle is half the area of the rectangle, or
bh.
2
This result is true of all triangles and is formally stated in Theorem 10-3.
h
b
Areas of Triangles and Trapezoids
Consider the area of this rectangle
Height
A(rectangle) = bh
Base
A(Triangle)
bh

2
Areas of Triangles and Trapezoids
If a triangle has an area of A square units, a base of b units,
and a corresponding altitude of h units, then
1
A  bh
2
Theorem
10-3
Area of a
Triangle
h
b
Areas of Triangles and Trapezoids
Find the area of each triangle:
4
1
yd
3
6 yd
A = 13 yd2
18 mi
23 mi
A = 207 mi2
Because
thewill
opposite
of aof
parallelogram
have the same
length,to first
Next we
look atsides
the area
trapezoids. However,
it is helpful
rectangle
the understand
area of a parallelogram
is closely related to the area of a ________.
parallelograms.
height
base
height
The area of a parallelogram is found by multiplying the base
____ and the ______.
Base – the bottom of a geometric figure.
Height – measured from top to bottom, perpendicular to the base.
Areas of Triangles and Trapezoids
Starting with a single trapezoid.
The height is labeled h, and the bases are labeled b1 and b2
Construct a congruent trapezoid and arrange it so that a pair of congruent legs
are adjacent.
b1
b2
h
b1
b2
The new, composite figure is a parallelogram.
It’s base is (b1 + b2) and it’s height is the same as the original trapezoid.
The area of the parallelogram is calculated by multiplying the base X height.
A(parallelogram) = h(b1 + b2)
The area of the trapezoid is one-half of the parallelogram’s area.
1
ATrapezoid  hb1  b2 
2
Areas of Triangles and Trapezoids
If a trapezoid has an area of A square units,
bases of b1 and b2 units, and an altitude of h units, then
1
A  hb1  b2 
2
Theorem
10-4
Area of a
Trapezoid
b1
h
b2
Areas of Triangles and Trapezoids
Find the area of the trapezoid:
20 in
18 in
38 in
A = 522 in2
Areas of Triangles and Trapezoids
Areas of Regular Polygons
You will learn to find the areas of regular polygons.
1) center
2) apothem
Areas of Regular Polygons
Every regular polygon has a ______,
center a point in the interior that is equidistant
from all the vertices.
A segment drawn from the center that is perpendicular to a side of the regular
apothem
polygon is called an ________.
congruent
In any regular polygon, all apothems are _________.
Areas of Regular Polygons
Now,
create
by
drawing
segments
from
the
center
each
vertex on
Now
The figure
multiply
shows
timesis
the
a
center
number
and
ofall
triangles
vertices
that
of
amake
regular
uptopentagon.
the
regular
area
ofbelow
athis
triangle
calculated
with
the
following
formula:
perpendicular to a side.
An apothem
drawn
from the center, and is _____________
either
side ofisthe
apothem.
polygon.
72
There are 5 vertices and each is 72° from the other (360 ÷ 5 = ___.)
5 1 sa
2
72° 72°
72°
72°
72°
a
s
What measure does 5s represent? perimeter
Rewrite the formula for the area of a pentagon using P for perimeter.
1
aP
2
Areas of Regular Polygons
If a regular polygon has an area of A square units, an
apothem of a units, and a perimeter of P units, then
1
A  aP
2
Theorem 10-5
Area of a
Regular
Polygon
P
Areas of Regular Polygons
Find the area of the shaded region in the regular polygon.
Area of polygon
A
1
aP
2
Area of triangle
A
1
bh
2
1
A  5.5 ft 40 ft 
2
1
A  5.5 ft 8 ft 
2
A  5.5 ft 20 ft 
A  5.5 ft 4 ft 
A  110 ft2
A  22 ft2
8 ft
5.5 ft
triangle
To find the area of the shaded region, subtract the area of the _______
pentagon
from the area of the ________:
The area of the shaded region:
110 ft2 – 22 ft2 =
88 ft2
Areas of Regular Polygons
Find the area of the shaded region in the regular polygon.
Area of polygon
A
1
aP
2
1
A  6.9m48 m
2
Area of triangle
A
1
bh
2
1
A  8m13 .8m
2
A  6.9m24 m
A  4m13.8m
A  165 .6m 2
A  55 .2m 2
6.9 m
triangle
To find the area of the shaded region, subtract the area of the _______
hexagon
from the area of the ________:
The area of the shaded region:
165.6 m2 – 55.2 m2 = 110.4 m2
8m
Areas of Regular Polygons
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