polygons and area
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Polygons and AreaPolygons and AreaPolygons and AreaPolygons and Area
§§ 10.1 Naming Polygons 10.1 Naming Polygons
§§ 10.4 Areas of Triangles and Trapezoids 10.4 Areas of Triangles and Trapezoids
§§ 10.3 Areas of Polygons 10.3 Areas of Polygons
§§ 10.2 Diagonals and Angle Measure 10.2 Diagonals and Angle Measure
§§ 10.6 Symmetry 10.6 Symmetry
§§ 10.5 Areas of Regular Polygons 10.5 Areas of Regular Polygons
§§ 10.7 Tessellations 10.7 Tessellations
Naming Polygons Naming Polygons
You will learn to name polygons according to the number of _____ and ______.
1) regular polygon
2) convex
3) concave
sides angles
Naming Polygons Naming Polygons
A polygon is a _____________ in a plane formed by segments, called sides.closed figure
A polygon is named by the number of its _____ or ______.sides angles
A triangle is a polygon with three sides. The prefix ___ means three.tri
Naming Polygons Naming Polygons
Prefixes are also used to name other polygons.
Prefix Number of
Sides
Name of
Polygon
tri-
quadri-
penta-
hexa-
hepta-
octa-
nona-
deca-
3
4
5
6
7
8
9
10
triangle
quadrilateral
pentagon
hexagon
heptagon
octagon
nonagon
decagon
Naming Polygons Naming Polygons
U
TS
Q
R
P
A vertex is the point of intersection of two sides.
A segment whoseendpoints arenonconsecutivevertices is adiagonal.
Consecutive vertices arethe two endpoints of anyside.
Sides that share a vertexare called consecutive sides.
Naming Polygons Naming Polygons
An equilateral polygon has all _____ congruent.
An equiangular polygon has all ______ congruent.
sides
angles
A regular polygon is both ___________ and ___________.equilateral equiangular
equilateralbut not
equiangular
equiangularbut not
equilateral
regular,both equilateraland equiangular
Investigation: As the number of sides of a series of regular polygons increases, what do younotice about the shape of the polygons?
Naming Polygons Naming Polygons
A polygon can also be classified as convex or concave.
If all of the diagonalslie in the interior of the figure, then the
polygon is ______.convex
If any part of a diagonal liesoutside of the figure, then thepolygon is _______.concave
Diagonals and Angle Measure Diagonals and Angle Measure
You will learn to find measures of interior and exterior anglesof polygons.
Nothing New!
Diagonals and Angle Measure Diagonals and Angle Measure
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
1) Draw a convex quadrilateral.
2) Choose one vertex and draw all possible diagonals from that vertex.
3) How many triangles are formed?
Make a table like the one below.
Diagonals and Angle Measure Diagonals and Angle Measure
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
1) Draw a convex pentagon.
2) Choose one vertex and draw all possible diagonals from that vertex.
3) How many triangles are formed?
pentagon 5 2 3 3(180) = 540
Diagonals and Angle Measure Diagonals and Angle Measure
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
1) Draw a convex hexagon.
2) Choose one vertex and draw all possible diagonals from that vertex.
3) How many triangles are formed?
pentagon 5 2 3 3(180) = 540
hexagon 6 3 4 4(180) = 720
Diagonals and Angle Measure Diagonals and Angle Measure
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
1) Draw a convex heptagon.
2) Choose one vertex and draw all possible diagonals from that vertex.
3) How many triangles are formed?
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 Diagonals and Angle Measure
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
1) Any convex polygon.
2) All possible diagonals from one vertex.
3) How many triangles?
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
Theorem 10-1If a convex polygon has n sides, then the sum of the measure of its interior angles is (n – 2)180.
Diagonals and Angle Measure Diagonals and Angle Measure
57°48°
74°
55°54°
72°
In §7.2 we identified exterior angles of triangles.
Likewise, you can extend the sides of anyconvex polygon to form exterior angles.
The figure suggests a method for finding thesum of the measures of the exterior angles of a convex polygon.
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 measures of linear pairs
sum of measures of interior angles=
=
–
–n•180 180(n – 2)= –180n 180n + 360
= 360sum of measure of exterior angles
Diagonals and Angle Measure Diagonals and Angle Measure
Theorem 10-2In any convex polygon, the sum of the measures of the
exterior angles, (one at each vertex), is 360.
Java Applet
Areas of Polygons 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 Areas of Polygons
Any polygon and its interior are called a ______________.polygonal region
In lesson 1-6, you found the areas of rectangles.
Postulate 10-1
Area Postulate
For any polygon and a given unit of measure, there is a unique number A called the measure of the area of the 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 Areas of Polygons
The figures above are examples of ________________.composite figures
They are each made from a rectangle and a triangle that have been placedtogether. You can use what you know about the pieces to gain informationabout the figure made from them.
You can find the area of any polygon by dividing the original region into smaller and simpler polygon regions, like _______, __________, and ________.
rectanglessquarestriangles
The area of the original polygonal region can then be found by ___________________________________.
adding theareas of the smaller polygons
Areas of Polygons Areas of Polygons
Postulate 10-3
Area Addition Postulate
The area of a given polygon equals the sum of the areas of the non-overlapping polygons that form the given polygon.
AreaTotal = A1 + A2 + A3
1
2 3
Areas of Polygons Areas of Polygons
3 units
3 units
Area of Square3u X 3u = 9u2
Area of Rectangle1u X 2u = 2u2
Area of RectangleArea of Square
Find the area of the polygon in square units.
Area of polygon =
= 7u2
Areas of Triangles and Trapeziods Areas of Triangles and Trapeziods
You will learn to find the areas of triangles and trapezoids.
Nothing new!
Areas of Triangles and Trapezoids Areas of Triangles and Trapezoids
b
h
Look at the rectangle below. Its area is bh square units.
The diagonal divides the rectangle into two _________________.congruent triangles
The area of each triangle is half the area of the rectangle, or .bh2
1
This result is true of all triangles and is formally stated in Theorem 10-3.
Areas of Triangles and Trapezoids Areas of Triangles and Trapezoids
Consider the area of this rectangle
A(rectangle) = bh
Base
Heig
ht
2
bhA Triangle )(
Areas of Triangles and Trapezoids Areas of Triangles and Trapezoids
Theorem
10-3
Area of a
Triangle
If a triangle has an area of A square units,
bhA2
1
b
h
a base of b units,
and a corresponding altitude of h units, then
Areas of Triangles and Trapezoids Areas of Triangles and Trapezoids
Find the area of each triangle:
A = 13 yd2
yd3
14
6 yd
18 mi
23 mi
A = 207 mi2
Because the opposite sides of a parallelogram have the same length,the area of a parallelogram is closely related to the area of a ________.rectangle
The area of a parallelogram is found by multiplying the ____ and the ______.base height
base
height
Base – the bottom of a geometric figure.Height – measured from top to bottom, perpendicular to the base.
Next we will look at the area of trapezoids. However, it is helpful to first understand parallelograms.
Areas of Triangles and Trapezoids Areas of Triangles and Trapezoids
h
b1
b2b1
b2
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 legsare adjacent.
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.
212
1bbhA Trapezoid
Areas of Triangles and Trapezoids Areas of Triangles and Trapezoids
Theorem
10-4
Area of a
Trapezoid
If a trapezoid has an area of A square units,
h
b1
b2
bases of b1 and b2 units, and an altitude of h units, then
212
1bbhA
Areas of Triangles and Trapezoids Areas of Triangles and Trapezoids
Find the area of the trapezoid:
A = 522 in2
18 in
20 in
38 in
Areas of Regular Polygons Areas of Regular Polygons
You will learn to find the areas of regular polygons.
1) center
2) apothem
Areas of Regular Polygons Areas of Regular Polygons
Every regular polygon has a ______, a point in the interior that is equidistantfrom all the vertices.
center
A segment drawn from the center that is perpendicular to a side of the regularpolygon is called an ________.apothem
In any regular polygon, all apothems are _________.congruent
Areas of Regular Polygons Areas of Regular Polygons
72° 72°
72°72°
72°
s
a
The figure below shows a center and all vertices of a regular pentagon.
There are 5 vertices and each is 72° from the other (360 ÷ 5 = ___.)72An apothem is drawn from the center, and is _____________ to a side.perpendicularNow, create a triangle by drawing segments from the center to each vertex oneither side of the apothem.The area of a triangle is calculated with the following formula:
sa2
1
Now multiply this times the number of triangles that make up the regularpolygon.
sa2
15
What measure does 5s represent? perimeter
Rewrite the formula for the area of a pentagon using P for perimeter. aP2
1
Areas of Regular Polygons Areas of Regular Polygons
Theorem 10-5
Area of a
Regular Polygon
If a regular polygon has an area of A square units, an
apothem of a units, and a perimeter of P units, then
aPA2
1
P
Areas of Regular Polygons Areas of Regular Polygons
5.5 ft
8 ft
Find the area of the shaded region in the regular polygon.
Area of polygon
aPA2
1
ftftA 40552
1.
ftftA 2055.
2110 ftA
Area of triangle
bhA2
1
ftftA 8552
1.
ftftA 455.
222 ftA
To find the area of the shaded region, subtract the area of the _______from the area of the ________:
trianglepentagon
The area of the shaded region: 110 ft2 – 22 ft2 = 88 ft2
Areas of Regular Polygons Areas of Regular Polygons
Find the area of the shaded region in the regular polygon.
Area of polygon
aPA2
1
mmA 48962
1.
mmA 2496.
26165 mA .
Area of triangle
bhA2
1
mmA 81382
1.
mmA 8134 .
2255 mA .
To find the area of the shaded region, subtract the area of the _______from the area of the ________:
trianglehexagon
The area of the shaded region: 165.6 m2 – 55.2 m2 = 110.4 m2
6.9 m 8 m
Areas of Regular Polygons Areas of Regular Polygons
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