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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. Vocabulary. Naming Polygons . What You'll Learn.

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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


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Vocabulary

Naming Polygons

What You'll Learn

You will learn to name polygons according to the number of

_____ and ______.

sides

angles

1) regular polygon

2) convex

3) concave


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Naming Polygons

closed figure

A polygon is a _____________ in a plane formed by segments, called sides.

sides

angles

A polygon is named by the number of its _____ or ______.

A triangle is a polygon with three sides. The prefix ___ means three.

tri


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Naming Polygons

Prefixes are also used to name other polygons.

tri-

3

triangle

quadri-

4

quadrilateral

penta-

5

pentagon

hexa-

6

hexagon

hepta-

7

heptagon

octa-

8

octagon

nona-

9

nonagon

deca-

10

decagon


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Q

P

R

U

S

T

Naming Polygons

Terms

Consecutive vertices are

the two endpoints of any

side.

A vertex is the point of intersection of

two sides.

A segment whose

endpoints are

nonconsecutive

vertices is a

diagonal.

Sides that share a vertex

are called consecutive

sides.


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equiangular

but not

equilateral

regular,

both equilateral

and equiangular

equilateral

but not

equiangular

Naming Polygons

sides

An equilateral polygon has all _____ congruent.

angles

An equiangular polygon has all ______ congruent.

equilateral

equiangular

A regular polygon is both ___________ and ___________.

Investigation: As the number of sides of a series of regular polygons increases, what do you

notice about the shape of the polygons?


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Naming Polygons

A polygon can also be classified as convex or concave.

If any part of a diagonal lies

outside of the figure, then the

polygon is _______.

If all of the diagonals

lie in the interior of

the figure, then the

polygon is ______.

concave

convex


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Naming Polygons

End of Section 10.1


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Vocabulary

Diagonals and Angle Measure

What You'll Learn

You will learn to find measures of interior and exterior angles

of polygons.

Nothing New!


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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?

1

4

2

quadrilateral

2(180) = 360


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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?

1

4

2

quadrilateral

2(180) = 360

2

5

3

pentagon

3(180) = 540


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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?

1

4

2

quadrilateral

2(180) = 360

2

5

3

pentagon

3(180) = 540

3

6

4

hexagon

4(180) = 720


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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?

1

4

2

quadrilateral

2(180) = 360

2

5

3

pentagon

3(180) = 540

3

6

4

hexagon

4(180) = 720

4

7

5

heptagon

5(180) = 900


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Diagonals and Angle Measure

1) Any convex polygon.

2) All possible diagonals from one vertex.

3) How many triangles?

1

4

2

quadrilateral

2(180) = 360

2

5

3

pentagon

3(180) = 540

3

6

4

hexagon

4(180) = 720

4

7

5

heptagon

5(180) = 900

n - 3

n

n - 2

n-gon

(n – 2)180


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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.

48°

57°

74°

The figure suggests a method for finding the

sum of the measures of the exterior anglesof a convex polygon.

72°

55°

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 measures of

linear pairs

sum of measures of

interior angles

=

=

n•180

180(n – 2)

=

180n

180n + 360

sum of measure of

exterior angles

=

360



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Diagonals and Angle Measure

End of Section 10.2


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Vocabulary

Areas of Polygons

What You'll Learn

You will learn to calculate and estimate the areas of polygons.

1) polygonal region

2) composite figure

3) irregular figure


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Areas of Polygons

polygonal region

Any polygon and its interior are called a ______________.

In lesson 1-6, you found the areas of rectangles.

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.


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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

smaller and simpler polygon regions, like _______, __________,

and ________.

rectangles

squares

triangles

adding the

The area of the original polygonal region can then be found by __________

_________________________.

areas of the smaller polygons


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1

2

3

Areas of Polygons

AreaTotal =

A1 +

A2 +

A3


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Area of Rectangle

1u X 2u = 2u2

Area of Square

3u X 3u = 9u2

3 units

3 units

Areas of Polygons

Find the area of the polygon in square units.

Area of polygon =

= 7u2

Area of

Rectangle

Area of Square


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Areas of Polygons

End of Section 10.3


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Vocabulary

Areas of Triangles and Trapezoids

What You'll Learn

You will learn to find the areas of triangles and trapezoids.

Nothing new!


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h

b

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 _________________.

The area of each triangle is half the area of the rectangle, or

This result is true of all triangles and is formally stated in Theorem 10-3.


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Height

Base

Areas of Triangles and Trapezoids

Consider the area of this rectangle

A(rectangle) = bh


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h

b

Areas of Triangles and Trapezoids

a base of b units,

and a corresponding altitude of h units, then


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18 mi

6 yd

23 mi

Areas of Triangles and Trapezoids

Find the area of each triangle:

A = 207 mi2

A = 13 yd2


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height

base

Because the opposite sides of a parallelogram have the same length,the area of a parallelogram is closely related to the area of a ________.

Next we will look at the area of trapezoids. However, it is helpful to first understand parallelograms.

rectangle

height

The area of a parallelogram is found by multiplying the ____ and the ______.

base

Base – the bottom of a geometric figure.

Height – measured from top to bottom, perpendicular to the base.


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b1

b2

h

b2

b1

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.

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.


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b1

h

b2

Areas of Triangles and Trapezoids

bases of b1 and b2 units,

and an altitude of h units, then


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20 in

18 in

38 in

Areas of Triangles and Trapezoids

Find the area of the trapezoid:

A = 522 in2



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Vocabulary

Areas of Regular Polygons

What You'll Learn

You will learn to find the areas of regular polygons.

1) center

2) apothem


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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

polygon is called an ________.

apothem

congruent

In any regular polygon, all apothems are _________.


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a

s

Areas of Regular Polygons

Now, create a triangle by drawing segments from the center to each vertex on

either side of the apothem.

Now multiply this times the number of triangles that make up the regularpolygon.

The figure below shows a center and all vertices of a regular pentagon.

The area of a triangle is calculated with the following formula:

perpendicular

An apothem is drawn from the center, and is _____________ to a side.

There are 5 vertices and each is 72° from the other (360 ÷ 5 = ___.)

72

72°

72°

72°

72°

72°

What measure does 5s represent?

perimeter

Rewrite the formula for the area of a pentagon using P for perimeter.



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8 ft

5.5 ft

Areas of Regular Polygons

Find the area of the shaded region in the regular polygon.

Area of polygon

Area of triangle

triangle

To find the area of the shaded region, subtract the area of the _______

from the area of the ________:

pentagon

The area of the shaded region:

88 ft2

110 ft2 – 22 ft2 =


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6.9 m

8 m

Areas of Regular Polygons

Find the area of the shaded region in the regular polygon.

Area of polygon

Area of triangle

triangle

To find the area of the shaded region, subtract the area of the _______

from the area of the ________:

hexagon

The area of the shaded region:

110.4 m2

165.6 m2 – 55.2 m2 =



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Vocabulary

Section Title

What You'll Learn

You will learn to

1)



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Vocabulary

Section Title

What You'll Learn

You will learn to

1)