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Mrs. Kummer

Fall, 2011-2012

10-1: Areas of Parallelograms & Triangles

Background:

Once you know what a

dimension does for you,

you can take two

dimensions and combine

them for the Area. This

is used in construction,

landscaping, home

improvement projects,

etc.

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10-1: Areas of Parallelograms & TrianglesVocabulary:

Dimension: Measurement of distance in one direction.

Area,A: Product of any 2 dimensions. Measures an object’s INTERIOR and has square units. Ex. m2, cm2, ft2

Volume, V: Product of any 3 dimensions. Measures an objects INTERIOR PLUS DEPTH and has cubed units. Ex. m3, cm3, ft3

Base: The side of any shape that naturally sits on the ground or any surface

Height: The side of any shape that is to base.

Parallelogram: A shape with 2 sets of parallel sides.

NOTE: SLANTED SIDES ≠ HEIGHT

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Ex.1 Label each side as a base or height or nothing.

a.

b.

c.

10-1: Areas of Parallelograms & Triangles

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97

9

7

8

7

12

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Now that you can identify the base and height properly,

now calculate the area of any shape. Use your formula

sheet for the various formulas for shapes.

Area of Parallelogram, Area of Triangle

Rectangle, or Square

A = b· h A = ½ · b ·h

A = s2 (for Square)

CH10-1: Areas of Parallelograms & Triangles

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h

b

h

b

CH10-1: Areas of Parallelograms & Triangles

How To Use It:

Ex.2 Find the area of each triangle, given the base b and the height h.

b = 8, h=2

A = ½∙(b∙h)

A = ½∙(8∙2)

A = 8

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Ex. 3 What is the area of DEF with vertices D(-1,-5),

E(4,-5) and F(4, 7)?

Plot it on x-y coordinate system

Connect dots.

Count how long b is

Count how long h is

Use Area of Formula.

A = ½∙(b∙h)

A = ½*(5∙12)

A = 30

10-1: Areas of Parallelograms & Triangles

F

D E

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Now, you do ODDS 1-19

10-1: Areas of Parallelograms & Triangles

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What about weird shapes like trapezoids or kites?

Kites/Rhombuses: Find area by finding the lengths of the

two diagonals and plug into formula.

Trapezoids: Find area by finding two bases and height

using trig. functions.

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10-2: Areas of Trapezoids,

Rhombuses, and Kites

diagonal 1, d1

diagonal 2, d2

b2

b1

h

A = ½ · d1 ·d2

Area of Kite A = ½ · h (b1 + b2)

Area of Trapezoid

Ex.1 Find the area of each kite.

A = ½d1∙d2

A = ½∙(9ft)(12ft)

A = 54 ft2

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10-2: Areas of Trapezoids,

Rhombuses, and Kites

6ft

9ft

6ft

Ex.1 Find the area of each trapezoid.

First, find h with trig. functions.

Tan(60°) = h/6.4

1.7321 = h

1 6.4

h = 11.1

A = ½h(b1+b2)

A= ½(11.1)(14.2 +20.6)

A= 193.14 in2

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10-2: Areas of Trapezoids,

Rhombuses, and Kites

6ft

14.2 in.

20.6 in

60°

Now, you do EVENS 2-14

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10-2: Areas of Trapezoids,

Rhombuses, and Kites

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10-3 Area of Regular Polygons Background: Not all shapes are triangles, rectangles, and

parallelograms. Think about your drive home: how

many different shapes exist in the street signs you see?

Vocabulary:

Polygon: any shape with 3 or more sides.

Center: the center of the imaginary circle that can be

made on the outside of the polygon.

Apothem: the height of the polygon. You find it by

making an isosceles triangle and using trig functions or

Pythagorean Theorem.

Central Angle (CA)°: angle made from center to any

vertex. CA° = 360°/n n = number of sides of polygon13

10-3 Area of Regular Polygons How To Use It:

Ex.1 Find the central angle of the following polygon.

n = 8

CA° = 360°

n

CA° = 360°

8

CA° =45°14

10-3 Area of Regular Polygons How To Use It:

Ex.2 Find the values of the variables for each regular

hexagon.

n = 6

CA° = 360°

n

CA° = 360°

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CA° =60° which is…which letter?

b°!

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4

db°c

10-3 Area of Regular Polygons How To Use It:

Ex.2 Find the values of the variables for each regular

hexagon.

To find c and d, you need

Trig functions.

First, bisect b°

b° becomes 30°

Now, go through trig recipe.

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d30°c

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10-3 Area of Regular Polygons How To Use It:

Tan (z°) = O

A

Tan (30°) = O

4

0.5774 = O

4

O = 2.31

But this is half of d, so

d = 4.62

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4

d30°c

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10-3 Area of Regular Polygons How To Use It:

Cos (z°) = A

H

Cos (30°) = 4

c

0.8660 = 4

1 c

0.8660c = 4

0.8660 0.8660

c = 4.62

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4

d30°c

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10-3 Area of Regular Polygons Vocabulary:

Area of a Polygon:

n

A = ½∙a∙n∙s

A = Area

a = apothem

n = number of sides

s = length of side

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a

s

10-3 Area of Regular Polygons Now, you try

ODDS 1 -11

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10-4 Perimeters and Areas of Similar Shapes

Background: Sometimes, you don’t have all the dimensions of all sides for your shapes. So, if you know the perimeters or areas, you can make a proportion to figure it out.

Vocabulary:

Perimeter: Sum of all sides of any shape. The “outside” dimension.

Area: The total amount of the “inside” of any shape.

Proportion: Two ratios set equal to each other.

Similarity Ratio: The ratio of two corresponding sides of two shapes, written as a/b

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A1 = a2

A2 b2

P1 = a

P2 b

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10-4 Perimeters and Areas of Similar Shapes

a

b

How To Use It:

Ex.1 For each pair of similar figures, find the ratios of the

perimeters and areas.

P1 = a A1 = a2

P2 b A2 b2

P1 = 4 A1 = 42

P2 3 A2 32

A1 = 16A2 9

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10-4 Perimeters and Areas of Similar Shapes

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4

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Now, you do EVENS

2, 4, and 6 in 10 minutes!

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10-4 Perimeters and Areas of Similar Shapes

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How To Use It:

Ex.2 For each pair of similar figures, the area of the smaller

shape is given. Find the missing area.

A1 = a2

A2 b2

50 = 32

A2 152

50(225) = A2 (9)

A2 = 1250 in225

10-4 Perimeters and Areas of Similar Shapes

A = 50 in2

3 in

15 in

Now, you do EVENS

8-14 in 15 minutes!

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10-4 Perimeters and Areas of Similar Shapes

10-5 Trigonometry and AreaBackground:

Sometimes, shortcuts can be made for special situations. For example, if you have a S-A-S triangle, you can use special area formulas to find the area.

Vocabulary:

S-A-S (Side-Angle-Side) Triangle: Two sides of a triangle are known, and the included angle is known.

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8

15

36°

10-5 Trigonometry and AreaRemember the area of a triangle is:

A = ½ ·b ·h

b = ?

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h = ?

Don’t know, need trig first to find it%.

Or a shortcut!

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10-5 Trigonometry and AreaArea of Triangle for S-A-S Triangle

A = ½ ·b · c · (sin A°)

Where b and c are sides of included angle

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c

b

A°

a

B

C

A

10-5 Trigonometry and AreaHow To Use It:

Ex.1 Find the area of each triangle.

Which formula do I use?

Easy…when it gives you an included angle, use the Area-Trig

formula!

First, label the triangle….30

8

15

36°

b

c

A°

10-5 Trigonometry and AreaNow, write formula once:

A = ½ ∙b ∙ c ∙ (sin A°)

A = ½ ∙8 ∙ 15 ∙ (sin 36°) Plug in what you know

A = 35.27 u2 Chug it on calculator

Don’t forget your units!!!!!!!!

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10-5 Trigonometry and Area

YOU DO 9-17

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CH 10-6 Circles and ArcsBackground: Circles have many measurements that can be

taken: circumference, lengths of arcs, areas, diameters,

and radii (plural for radius).

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d

CH 10-6 Circles and ArcsVocabulary:

Circumference: Sum of the outside. C = π∙d

Major arc: Distance GREATER than half of the circle

Minor arc: Distance LESS than half of the circle

Semicircle: Distance of half of the circle

Measure of an arc (°): Central angles sum to 360°, and semicircle arcs

measure 180 °

Length of an arc (cm, m, in): arc (°) ∙2∙π∙r

360(°)

Diameter: a measure from end to end of a circle, passing through the

center.

Radius: Half of the diameter34

CH 10-6 Circles and ArcsHow To Use It:

Ex. 1: Find the circumference of each side. Leave your answers in terms of π.

r=12, so

d=24

C = π∙d

C = π∙24

C = 24π

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Now, you do all,

1-3 in 5 minutes!

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10-6 Circle and Arcs

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CH 10-6 Circles and ArcsHow To Use It:

Ex.2 State whether the following is a minor or major arc.

DB

Minor arc

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DA

B

C

CH 10-6 Circles and Arcs

Now, you do

4-14

in 10 minutes!38

CH 10-6 Circles and ArcsHow To Use It:

Ex.3 Find the measure of each arc in the circle.

DAB °=?

ACD = 180°

AB = 180°-70°

AB = 110°

DAB = ACD + AB

DAB = 290°39

DA

B

C

70°

CH 10-6 Circles and ArcsNow you do

15-20

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CH 10-6 Circles and ArcsEx. 4 Find the length of each arc.

BD = ?

Length BD = (mBD) ∙2∙π∙r

(360)

Length BD = (90) ∙2∙π∙13

(360)

Length = 0.25∙26 ∙π

BD = 6.5 π

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DA

B

26 in

CH 10-6 Circles and Arcs

Now you do

21,22,and23

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CH 10-7 Areas of Circles and SectorsVocabulary:

Area of a Circle: A = π∙r2

Area of a Sector of a Circle: Asector = arc (°) ∙π∙r2

360(°)

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CH 10-7 Areas of Circles and SectorsEx. 1 Find the area of the shaded segment. Leave your

answer in terms of π

Areasector = (mBD) ∙π∙r2

(360)

Areasector= (90) ∙π∙82

(360)

Areasector = 0.25 ∙π∙64

Areasector = 16π in2

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DA

B

8 in

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CH 10-7 Areas of Circles and Sectors

Now you do ODDS

1-19

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YAHOO!!!!!!!

We’re done with CH10!

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