§ 10.1 Naming Polygons Naming PolygonsNaming Polygons § 10.4 Areas of Triangles and Trapezoids § 10.3 Areas of Polygons Areas of PolygonsAreas of

Polygons and AreaPolygons and AreaPolygons and AreaPolygons and Area

§§ 10.1 10.1 Naming Polygons

§§ 10.4 Areas of Triangles and Trapezoids 10.4 Areas of Triangles and Trapezoids

§§ 10.3 10.3 Areas of Polygons

§§ 10.2 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


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


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


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.


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?

http://robertfant.com/Java/AreaCircle.htm


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!


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.


Convex

Polygon

Number

of Sides


Number of

Triangles

Sum of

Interior Angles


1) Draw a convex pentagon.



pentagon 5 2 3 3(180) = 540


Convex

Polygon

Number

of Sides


Number of

Triangles

Sum of

Interior Angles


1) Draw a convex hexagon.



pentagon 5 2 3 3(180) = 540

hexagon 6 3 4 4(180) = 720


Convex

Polygon

Number

of Sides


Number of

Triangles

Sum of

Interior Angles


1) Draw a convex heptagon.



pentagon 5 2 3 3(180) = 540

hexagon 6 3 4 4(180) = 720

heptagon 7 4 5 5(180) = 900


Convex

Polygon

Number

of Sides


Number of

Triangles

Sum of

Interior Angles


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.


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


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


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.


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


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


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 Trapezoids Areas of Triangles and Trapezoids

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

Nothing new!


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.


Consider the area of this rectangle

A(rectangle) = bh

Base

Heig

ht

2

bhA Triangle )(


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


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.


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


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


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


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


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


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


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


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


Section Title Section Title

You will learn to

1)



You will learn to

1)


Documents

§ 10.1 Naming Polygons Naming PolygonsNaming Polygons § 10.4 Areas of Triangles and Trapezoids § 10.3 Areas of Polygons Areas of PolygonsAreas of