Ms. EllmerWinter, 2010-2011

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 surfaceHeight: The side of any shape that is to base.Parallelogram: A shape with 2 sets of parallel sides.

NOTE: SLANTED SIDES ≠ HEIGHT3

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

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7

8

7

124

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.

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

b = 8, h=2A = ½ (b h)∙ ∙A = ½ (8 2)∙ ∙A = 8

10-1: Areas of Parallelograms & Triangles

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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 systemConnect dots.Count how long b isCount how long h isUse 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 (skip 11)

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

Ex.1 Find the area of each kite.

A = ½d1 d∙ 2

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/6.41.7321 = h

1 6.4h = 11.1A = 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

10-5 Trigonometry and AreaYOU DO ODDS 1-17

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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 polygon

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

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

n = 8CA° = 360°

nCA° = 360°

8CA° =45°

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

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

n = 6CA° = 360°

nCA° = 360°

6CA° =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 needTrig functions.First, bisect b°b° becomes 30°Now, go through trig recipe.

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d30°c4

10-3 Area of Regular Polygons How To Use It:Tan (z°) = O

ATan (30°) = O

40.5774 = O

4O = 2.31But this is half of d, sod = 4.62

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d30°c4

10-3 Area of Regular Polygons How To Use It:Cos (z°) = A

HCos (30°) = 4

c0.8660 = 4

1 c0.8660c = 40.8660 0.8660c = 4.62

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4

d30°c4

10-3 Area of Regular Polygons Vocabulary: Area of a Polygon:

n

A = ½ a n s∙ ∙ ∙A = Areaa = apothemn = number of sidess = 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 ShapesBackground: 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.

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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 23

10-4 Perimeters and Areas of Similar Shapes

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4

4

Now, you do EVENS

2, 4, and 6 in 10 minutes!

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10-4 Perimeters and Areas of Similar Shapes

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 in15 in

Now, you do EVENS

8-14 in 15 minutes!

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10-4 Perimeters and Areas of Similar Shapes

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∙dMajor arc: Distance GREATER than half of the circleMinor arc: Distance LESS than half of the circleSemicircle: Distance of half of the circleMeasure 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 diameter 28

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, sod=24C = π∙dC = π∙24C = 24π

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Now, you do all,

1-3 in 5 minutes!

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10-6 Circle and Arcs

CH 10-6 Circles and ArcsHow To Use It:

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

BCDMinor arc

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D A

B

C

CH 10-6 Circles and ArcsNow, you do

4-9in 5 minutes!

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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 + ABDAB = 290°

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D A

B

C70°

CH 10-6 Circles and ArcsNow you do

16,18,20

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

BD = ?Length BD = mBD 2∙ ∙π∙r

360Length BD = 90 2∙ ∙π∙13

360Length = 0.25 26 ∙ ∙πBD = 6.5 π

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D A

B

26 in

CH 10-6 Circles and ArcsNow 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

360Areasector= 90 ∙π∙82

360Areasector = 0.25 ∙π∙64Areasector = 16π

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D A

B

8 in

CH 10-7 Areas of Circles and SectorsNow you do ODDS

9-17

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YAHOO!!!!!!!

We’re done with CH10!

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