Ms. Ellmer Winter, 2010-2011. 10-1: Areas of Parallelograms & Triangles Background: Once you know...
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Transcript of Ms. Ellmer Winter, 2010-2011. 10-1: Areas of Parallelograms & Triangles Background: Once you know...
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
9
7
8
7
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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.
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
6
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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4
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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4
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
3
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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