finding - edl · finding area can i remember and use formulas for the area of squares, rectangles,...

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


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


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


A = 70


4

February 08, 2012

8

5x


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?


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


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!


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


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


3

February 08, 2012


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


So the area of the polygon is half the apothem

times the side times the number of sides.


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 =


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


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


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?


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


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


.12

.10



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


4

February 08, 2012

Circumference of a Circle

a

C = πd

rd

C = 2πr

Find the circumference of this circle:

20


10


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


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=


6

February 08, 2012


.6

120o

X

Find the value of y:

.

20

90o

Y


.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?


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:


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 =


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


4

February 08, 2012

Can I find the area of

special quadrilaterls?

Can I memorize the formulas for the areas

of special quadrilaterals?


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


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


2

February 08, 2012

wanttotal possible

Wanted Length

Total Length=


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=


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


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=


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?


5

February 08, 2012

What is the probability of rolling doubles?

What is the probability of rolling 7 on these dice?


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.


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.


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


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


5

February 08, 2012


1

9

8

7

6

54

32


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=


For theoretical probability, we look at the central angles.

Since each color has a 90o central angle, they each have a 25% probability.


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.

..

It looks like:

a ring, or

a doughnut


7

February 08, 2012

..r

R

Area = πR2 - πr2

finding - edl · finding area can i remember and use formulas for the area of squares, rectangles,...

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