Transcript Geometry B CH10 AREA - Southgate Community School District

Ms. Ellmer
Winter, 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 & Triangles
Vocabulary:
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
3
10-1: Areas of Parallelograms & Triangles
Ex.1 Label each side as a base or height or nothing.
a.
7
9
8
b.
9
8
7
c.
7
12
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10-1: Areas of Parallelograms & Triangles
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=2
A = ½∙(b∙h)
A = ½∙(8∙2)
A=8
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10-1: Areas of Parallelograms & Triangles
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
F
Count how long h is
Use Area of Formula.
A = ½∙(b∙h)
A = ½*(5∙12)
E
A = 30
D
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10-1: Areas of Parallelograms & Triangles
Now, you do ODDS 1-19
(skip 11)
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10-2: Areas of Trapezoids,
Rhombuses, and Kites
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.
b2
diagonal 1, d1
h
diagonal 2, d2
b1
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10-2: Areas of Trapezoids,
Rhombuses, and Kites
Ex.1 Find the area of each kite.
A = ½d1∙d2
A = ½∙(9ft)(12ft)
A = 54 ft2
9ft
6ft
6ft
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10-2: Areas of Trapezoids,
Rhombuses, and Kites
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
20.6 in
60°
6ft
14.2 in.
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10-2: Areas of Trapezoids,
Rhombuses, and Kites
Now, you do EVENS 2-14
11
10-5 Trigonometry and Area
YOU DO ODDS 1-17
12
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=8
CA° = 360°
n
CA° = 360°
8
CA° =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=6
4
CA° = 360°
b°
n
c
d
CA° = 360°
6
CA° =60° which is…which letter?
b°!
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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.
To find c and d, you need
4
Trig functions.
First, bisect b°
c
30°
d
4
b° becomes 30°
Now, go through trig recipe.
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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
4
c
30°
4
d
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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
4
c
30°
4
d
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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
a
s
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10-3 Area of Regular Polygons
Now, you try
ODDS 1 -11
20
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.
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10-4 Perimeters and Areas of Similar Shapes
A1 = a2
A 2 b2
P1 = a
P2 b
a
b
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10-4 Perimeters and Areas of Similar Shapes
How To Use It:
Ex.1 For each pair of similar figures, find the ratios of the
perimeters and areas.
3
P1 = a
A 1 = a2
P2 b
A 2 b2
4
P1 = 4
P2 3
A1 = 42
A2 32
4
A1 = 16
A2 9
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10-4 Perimeters and Areas of Similar Shapes
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.
A 1 = a2
A 2 b2
50 = 32
A2 152
3 in
15 in
A = 50 in2
50(225) = A2 (9)
A2 = 1250 in2
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10-4 Perimeters and Areas of Similar Shapes
Now, you do EVENS
8-14 in 15 minutes!
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CH 10-6 Circles and Arcs
Background: Circles have many measurements that can be
taken: circumference, lengths of arcs, areas, diameters,
and radii (plural for radius).
d
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CH 10-6 Circles and Arcs
Vocabulary:
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 diameter
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CH 10-6 Circles and Arcs
How 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
12
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10-6 Circle and Arcs
Now, you do all,
1-3 in 5 minutes!
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CH 10-6 Circles and Arcs
How To Use It:
Ex.2 State whether the following is a minor or major
arc.
B
BCD
Minor arc
A
C
D
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CH 10-6 Circles and Arcs
Now, you do
4-9
in 5 minutes!
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CH 10-6 Circles and Arcs
How To Use It:
Ex.3 Find the measure of each arc in the circle.
B
DAB °=?
ACD = 180°
AB = 180°-70°
AB = 110°
DAB = ACD + AB
DAB = 290°
A
70°
C
D
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CH 10-6 Circles and Arcs
Now you do
16,18,20
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CH 10-6 Circles and Arcs
Ex. 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
A
B
26 in
D
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CH 10-6 Circles and Arcs
Now you do
21,22,and23
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CH 10-7 Areas of Circles and Sectors
Vocabulary:
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 Sectors
Ex. 1 Find the area of the shaded segment. Leave your
answer in terms of π
B
Areasector = mBD ∙π∙r2
360
Areasector= 90 ∙π∙82
360
Areasector = 0.25 ∙π∙64
Areasector = 16π
A
8 in
D
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CH 10-7 Areas of Circles and Sectors
Now you do ODDS
9-17
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YAHOO!!!!!!!
We’re done with CH10!
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