Geometry T1.notebook 1 February 08, 2012 Finding Area Can I remember and use formulas for the area of squares, rectangles, parallelograms, triangles, and irregular polygons? What is Area? It is the total of square units needed to fill a figure. We always multiply! Examples: carpet, paint, lawn, & roof. S S S S Area of a square: A square = s 2 Is A = wl? A SQ =s 2 Find the area of these squares: 8 10 Find the area of these squares: 3 √2 2x + 1 3x 1
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Geometry T1.notebook
1
February 08, 2012
FindingArea
Can I remember and useformulas for the area of
squares, rectangles, parallelograms, triangles, and irregular polygons?
What is Area?
It is the total of square units needed to fill a figure.
We always multiply!
Examples:carpet, paint, lawn, & roof.
S
S
SS
Area of a square:
Asquare = s2
Is A = wl?
ASQ = s2
Find the area of these squares:
8
10
Find the area of these squares:
3√2
2x + 1
3x 1
Geometry T1.notebook
2
February 08, 2012
Area of a rectangle:
WL
WL
Arectangle = WL
height (h)
base (b)
The area of a rectangle is the product of its base and height.
Arect. = bh
For example, find the areaof these rectangles:
8
9
5
2
Find the area of these rectangles: Find the value of x:
11
x A = 33
Find the value of x:
3x
4x
A = 192
Area Congruence
If two polygons are congruent, they have the same area
Geometry T1.notebook
3
February 08, 2012
The area of a parallelogram is the product of its base and height.
Apara. = bh
base
bx x
h
The area of a triangle is
half the product of its base and height.
Atriangle = bh/2
b
h
Area of a triangle:
h
a
b
Atriangle = bh/2
b
4
3
4
10
Find the area of these triangles:
67
Find the area of this triangle:
8
910x
20
Find the value of x:
A = 70
Geometry T1.notebook
4
February 08, 2012
8
5x
Find the value of x:
A = 100
Area Addition Postulate
The area of a region is the sum of its
nonoverlapping parts.
Find the area of this irregular polygon.
12
20
4
5
A = 128
6
11
9
4
2
A = 81
Find the area of this irregular polygon.
Can I remember and useformulas for the area of
squares, rectangles, parallelograms, triangles, and irregular polygons?
Geometry T2.notebook
1
February 08, 2012
Area of Regular Polygons
&Perimeter
Can I identify the parts of regular polygons?
Can I find the area of any regular polygon?
Can I find perimeter?
Terms that apply to regular polygons Every regular polygon can be inscribed in a circle!
.
.
..
..
So every regular polygon has a center
.
.
..
..
So every regular polygon has a radius
.
.
..
..
The radius goes from the center to a vertex
Geometry T2.notebook
2
February 08, 2012
Every regular polygon has a central angle
.
.
..
..
The central angle is made up of two radii
Every regular polygon has an apothem
.
.
..
..
The apothem is the distance from the center to a side
Match
center of circumscribed circle
center to side
vertex to center
apothem
polygon center
radius
Our next challenge: Find the area of any regular polygon.
Until today, we did not have formulas
for any regular polygon
and they must be REGULAR!
Our next challenge: Find the area of any regular polygon.
Note that all regular polygons are made up of n congruent triangles (made up of the n radii).
This hexagon has six congruent triangles
This decagon has ten congruent triangles
Our next challenge: Find the area of any regular polygon.
If we can find the area of one of the triangles, all we have to do is multiply that answer by n
for the polygon
Geometry T2.notebook
3
February 08, 2012
Our next challenge: Find the area of any regular polygon.
So lets look at just one
at just this one triangle
Our next challenge: Find the area of any regular polygon.If we add the apothem,we can find the area
The area of the triangle is: half of the apothem times the side
Our next challenge: Find the area of any regular polygon.
So the area of the polygon is half the apothem
times the side times the number of sides.
Our next challenge: Find the area of any regular polygon.
or
AREA =ans 2
AREA =ap 2
Try these problems:
Each apothem is 14Each side is 9
Area =
Try this:
Each side is 10The apothem = 9
Area =
Geometry T2.notebook
4
February 08, 2012
Try this:
The side is 2The apothem is 1
Area =
Try this:
Each side is 8 inches
Area =
Perimeter
It is the length it takes to go around the edge of a figure.
Examples:fences, frames, baseboards,
& outlines.
Perimeter of a square:
S
S
S
S
Psquare = 4s
Find the perimeter of this square:
8
Find the perimeter of this square:
11
Geometry T2.notebook
5
February 08, 2012
Perimeter of a rectangle:
W
L
W
L
Prectangle = 2L + 2W
Find the perimeter of this rectangle:
8
2
Find the perimeter of this rectangle:
9
5
Find x:
3x
2x
P = 130
Perimeter of a triangle:
c
ab
Ptriangle = a + b + c
Find the perimeter of this triangle:
5
12
13
Geometry T2.notebook
6
February 08, 2012
Find the perimeter of this triangle:
20
25
35
Find x:
5x 1
x + 10
3x + 7
P = 33
Can I identify the parts of regular polygons?
Can I find the area of any regular polygon?
Can I find perimeter?
Geometry T3.notebook
1
February 08, 2012
Circle Area, Arc Length, &
Circumference
Can I find the area of a circle?
Can I find the arc length of a circle?
Can I find the circumference of a circle?
What is the area of a circle?
One way to find out is to look at what we learned
the other day about regular polygons
.
.
.
..
.
The larger the number of sides, the closer their areas get to a circle!
With some computer help, we know that A = ap/2. If r = 1:
Getting close
By increasing the number of sideswe get closer, faster
We can get as close as we want to piby taking the right number of sides
Geometry T3.notebook
2
February 08, 2012
If the radius is one, the area is pi!
What if we use other radii?
Area of a Circle is πr2
Acircle = πr2
Area of a Circle
r
A = πr2
Try this: Find the area.
. 6
Try this: Find the area.
.12
.10
Try this: Find the area.
Geometry T3.notebook
3
February 08, 2012
.21
Try this: Find the area. Find the value of x:
.
Area = 25π
X
.Y
Area = 81π
Find the value of y: What is the Perimeter of a Circle?
Circles do not have perimeters, they have a circumference
(which is basically the same thing).
.
Circumference is the length around a circle.
C = πd
Since π = C/d
Since d = 2r
C = 2πr too
Geometry T3.notebook
4
February 08, 2012
Circumference of a Circle
a
C = πd
rd
C = 2πr
Find the circumference of this circle:
20
Find the circumference of this circle:
10
Find the circumference of this circle:
3
Try these:
If the radius is 12, the circumference is:
If the diameter is 6, the circumference is:
If the circumference is 9π, what is the diameter?
If the circumference is 8π, what is the radius?
What is arc length?
It is not arc measurement, it is the length of the arc.
..
A B
Geometry T3.notebook
5
February 08, 2012
..
A B
Our challenge:
Find the length of AB
)
..
A B
Find the central angle
..
A B
Notice that the arc is a part of the entire circumference
It is the same proportion as the central angle is to 360o
..
A B
..
That gives us a proportion as our formula
arc length
circumference
central angle
360o=
..
A B
..
Our challenge: Find the length of AB
)
Learn it!
This can be used to find arc length, circumference, or the central angle (arc measurement)
arc length
circumference
central angle
360o=
Geometry T3.notebook
6
February 08, 2012
Find the value of x:
.6
120o
X
Find the value of y:
.
20
90o
Y
Find the value of x:
.50
60o
X
Find the value of y:
.50
60o
Y
Find the value of z:
.
6π
135oZ
Can I find the area of a circle?
Can I find the arc length of a circle?
Can I find the circumference of a circle?
Geometry T4.notebook
1
February 08, 2012
Area of Special Quadrilaterals
Can I find the area of
special quadrilaterals?
Can I memorize the formulas for the areas
of special quadrilaterals?
The area of a trapezoid is the sum of its two triangles.
b2
b1
h
Atrap. = (b1h/2 + b2h/2)
The area of a trapezoid is half the product of the height
and the sum of its bases.
h(b1 + b2)/2
Atrap. = (b1 + b2)h/2
A = 8
16
1110
12Try this:
1820255A =
Try this:
Geometry T4.notebook
2
February 08, 2012
x
11
30
6
A = 150Find x: The area of a kite is
half the product of the diagonals.
d1
d2
Akite = d1d2/2
The area of a kite is the sum of two triangles.
d1d2a
d2b
Akite = d1d2/2
42
18
4
Try this:
A =
20
Find x:
x
A = 250
3
15
Try this:
5
A =
Geometry T4.notebook
3
February 08, 2012
The area of a rhombus is half the product of the diagonals.
Arhombus = d1d2/2
d1d2
The area of a rhombus is the sum of its two triangles.
Arhombus = d1d2/2
d1
d2a
d2b
15
Try this:
5
A =
5
Find x:
x
A = 80
8
Try these:
4
7
A = 4
10
5 A =
15
20
10A =
5
10.5
6 A = 3A =
Asquare = s2Arectangle = bh
Aparallelogram = bhAtriangle = bh/2
Atrapezoid = (b1 + b2)h/2Arhombus = d1d2/2
Akite = d1d2/2
Geometry T4.notebook
4
February 08, 2012
Can I find the area of
special quadrilaterls?
Can I memorize the formulas for the areas
of special quadrilaterals?
Geometry T5.notebook
1
February 08, 2012
GeometricProbability Can I define probability?
Can I apply probability to problems with length,
area, or angle?
What is probability?
Probability is a ratio of what you want to the total possible.
want:possible want %wanttotal possible
Example: If a single digit positive integer is selected at random, what is the probability that it is even?
1
9
8
7
6
54
32
Example: If a single digit positive integer is selected at random, what is the probability that it is even?
1
9
8
7
6
54
32
=4want
total possible 9
Example of linear probability:Find the probability that a point chosen at random between R and S is also between T and U.
10 2 3 4 5 6 7 8 9 1012345678910
R UT S
Geometry T5.notebook
2
February 08, 2012
wanttotal possible
Wanted Length
Total Length=
Example of linear probability:Find the probability that a point chosen at random between R and S is also between T and U.
10 2 3 4 5 6 7 8 9 1012345678910
R UT S
Wanted Length
Total Length = R to S = 10 0 = 10
T to U = 5 3 = 2 = 1
5
Try another: Find the probability that a point chosen at random between R and S is also between T and U.
10 2 3 4 5 6 7 8 9 1012345678910
R UT S
Try another: Find the probability that a point chosen at random between R and S is also between T and U.
10 2 3 4 5 6 7 8 9 1012345678910
R
UT
S
Example of area probability:Find the probability that a point chosen at random within the rectangle is also within the right triangle.
5
8
16
7
wanttotal possible
Wanted Area
Total Area=
Geometry T5.notebook
3
February 08, 2012
Example of area probability: Find the probability that a point chosen at random within the rectangle is also within the right triangle.
5
8
16
7
Wanted Area
Total Area = Triangle = 8x5/2 = 20
Rectangle = 16x7 = 112 =528
Another example of area probability: Find the probability that a point chosen at random within the circle is also within the square.
12
Another example of area probability: Find the probability that a point chosen at random within the circle is also within the equilateral triangle.
10
Example of angle probability:Find the probability that the spinner will end up in blue.
One way is to experiment:
Try it several times and keep track!
trys blue
Example of angle probability:Find the probability that the spinner will end up in blue.
For theoretical probability, we look at the central angles.
Since each color has a 90o central angle, they each have a 25% probability.
wantTotal Possible
Wanted Angle
Total Angle=
Geometry T5.notebook
4
February 08, 2012
Another example of angle probability:Find the probability that the spinner will end up X.
.X
Y
Another example of angle probability: Find the probability that the spinner will end up in C, D, or E.
.A B
90o
60o
30o
60o
C
DE
.A B
Another example of angle probability:Find the probability that the spinner will end up in A.
90o
60o
30o
60o
C
DE
Probability problems that involve length,
angle, or areaare geometric probility
problems
Can I define probability?
Can I apply probability to problems with length,
area, or angle?
What is the probability of rolling 3 on the blue die?
Geometry T5.notebook
5
February 08, 2012
What is the probability of rolling doubles?
What is the probability of rolling 7 on these dice?
Geometry T6.notebook
1
February 08, 2012
Unit T Review
S
S
SS
Area of a square:
Asquare = s2
Is A = wl?
ASQ = s2
height (h)
base (b)
The area of a rectangle is the product of its base and height.
Arect. = bh
The area of a parallelogram is the product of its base and height.
Apara. = bh
base
bx x
h
Area of a triangle:
h
a
b
Atriangle = bh/2
b
Area Addition Postulate
The area of a region is the sum of its
nonoverlapping parts.
Geometry T6.notebook
2
February 08, 2012
So every regular polygon has a center
.
.
..
..
So every regular polygon has a radius
.
.
..
..
The radius goes from the center to a vertex
Every regular polygon has a central angle
.
.
..
..
The central angle is made up of two radii
Every regular polygon has an apothem
.
.
..
..
The apothem is the distance from the center to a side
or
AREA =ans 2
AREA =ap 2
Perimeter
It is the length it takes to go around the edge of a figure.
Examples:fences, frames, baseboards,
& outlines.
Geometry T6.notebook
3
February 08, 2012
Perimeter of a square:
S
S
S
S
Psquare = 4s
Perimeter of a rectangle:
W
L
W
L
Prectangle = 2L + 2W
Perimeter of a triangle:
c
ab
Ptriangle = a + b + c Acircle = πr2
Area of a Circle
r
A = πr2
.
Circumference is the length around a circle.
Circumference of a Circle
a
C = πd
rd
C = 2πr
Geometry T6.notebook
4
February 08, 2012
arc length
circumference
central angle
360o=
..
A B
..
The area of a trapezoid is half the product of the height and the average of its bases.
h(b1 + b2)/2
Atrap. = (b1 + b2)h/2
The area of a kite is half the product of the diagonals.
d1
d2
Akite = d1d2/2
The area of a rhombus is half the product of the diagonals.
Arhombus = d1d2/2
d1d2
Asquare = s2Arectangle = bh
Aparallelogram = bhAtriangle = bh/2
Atrapezoid = (b1 + b2)h/2Arhombus = d1d2/2
Akite = d1d2/2
What is probability?
Probability is a ratio of what you want to the total possible.
want:possible want %wanttotal possible
Geometry T6.notebook
5
February 08, 2012
Example: If a single digit positive integer is selected at random, what is the probability that it is even?
1
9
8
7
6
54
32
Example of linear probability:Find the probability that a point chosen at random between R and S is also between T and U.
10 2 3 4 5 6 7 8 9 1012345678910
R UT S
wanttotal possible
Wanted Length
Total Length=
Example of area probability:Find the probability that a point chosen at random within the rectangle is also within the right triangle.
5
8
16
7
wanttotal possible
Wanted Area
Total Area=
Example of angle probability:Find the probability that the spinner will end up in blue.
For theoretical probability, we look at the central angles.
Since each color has a 90o central angle, they each have a 25% probability.
Geometry T6.notebook
6
February 08, 2012
wantTotal Possible
Wanted Angle
Total Angle= Honors Review
Area = s2√3 4 ..
A sector is the part of the interior of a circle between two radii and their intercepted arc.
....
Area of Sector
Area of circle
Central Angle
360o=
An annulus is the area between two concentric circles.