Area of Polygons

1
Chapter 11

• Area of Polygons

2
11.1

• Area of Parallelograms

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Area of Parallelograms
b
Lets take this parallelogram We know opposite
sides and opposite angles are congruent.
h
Drop an altitude from one vertex.
We have a rectangle andfrom elementary
schoolyou remember the area formula for a
rectangle isA length x width. Were going to
use A base x height.
Cut this triangle off the left sideand place on
the right side of the parallelogram.
What quadrilateral do we havenow?
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Area of Parallelograms
b
Base the base is oneof the sides of a
parathat has a height perpendicular to.
h
A b x h
Height the height of the parallelogram is the
distance between the parallel sides that are
the bases.
This formula works for ALL parallelograms!
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Properties of Area

• Areas can be added, subtracted, multiplied or
  divided.
• Added If you wanted to find the area of the
  walls of this room, you can find the area of each
  wall, then add them together.
• Subtracted If you wanted to find the area of
  the wall w/o the windows, you can find the area
  of the wall and subtract the window.

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Properties (Cont)

• Multiplied If you wanted to find the area of
  the wall, you can find the area of one of the
  cinder blocks then multiply by the number of
  blocks that are in the wall. (assuming theyre
  the same size)
• Divide If you knew the area of the wall and the
  number of blocks on the wall, you can divide them
  to find the area of each block (assuming theyre
  the same size).

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Example
Find the area of this figure.
Divide the figure into two rectangles.
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Rectangle I A 9 in x 2 in A 18 in2
I
8
9
Rectangle II A 8 in x 1 in A 8 in2
II
1
Total Area A 18 in2 8 in2 26 in2
10
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Example
Find the area of this figure.
What if we think of this as Area of Big
Rectangle minusArea of Little Rectangle, will
this work?
2
8
9
Area of Big (9)(10) 90
8
Area of Little (8)(8) 64
1
Total Area 90 64 26 in2
10
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11.2

• Area of Triangles, Trapezoids, Rhombi and Kites

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Area of Triangles

• Remember one of the characteristics of a
  parallelogram was that each diagonal divides the
  parallelogram in to two congruent triangles?
• If the two triangles are congruent, do they have
  the same area?
• So, if we knew the area of the parallelogram,
  then we can take ½ and find the area of a
  triangle.

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Area of Triangles
Divide the parallelograminto two congruent
triangles.
h
Base the base is one of the sides of the
triangle that has a height perpendicular to.
b
Apara b x h
Height the height of the triangle is the
distance between the vertex and the base.
A? (½) b x h
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Three Triangles One Formula
h
A? (½) b x h
b
A? (½) b x h
h
b
A? (½) b x h
h
b
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More Formulas for Triangles
Equilateral Triangle
s
s
s
Any Triangle (SSS Case)
b
a
(Where s is semi-perimeter and a, b and c are
sides)
c
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More Area Formulas
Any Triangle (SAS Case)
a
C
b
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Area of Quadrilaterals
This quadrilateral is not a parallelogram,
trapezoid, or a kite.
h
h
b
I
II
Divide the quadrilateral into two triangles and
find each area and add.
Area ?I (1/2)bh
Area ?II (1/2)bh
Area of Quad Area ?I Area ?II
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Area of Trapezoids
Draw a Median
h
Draw two altitudes through MPs of legs.
b
Cut the two trianglesfrom the bottom, rotateand
replace them on thetop.
We now have a rectangle wherethe base is the
median and the height is just the height.
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Area of Trapezoids
b2
h
b1
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Area of Rhombi and Kites
Area (1/2)(d1)(d2)
d2
d1
Also works for Squares b/c Squares are Rhombi
too.
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11.3

• Area of Regular Polygons and Circles

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Calculate Area
Draw an Regular Triangle
Find the Center
Draw segments from the center to each vertex.
We have created 3 congruent, Isosceles ?s.
h
Calculate the area of each little triangle, then
mult by 3.
b
A 3 (1/2)(bh)
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Area Continued
We just calculated the area as A 3 (1/2)(bh)
But 3b is the same asperimeter A (1/2)(Ph)
h
a
b
and h has a special name, Apothem (a), then
A (1/2)(Pa)
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Harder Example
A (1/2)Pa
(1/2) central angle
A (1/2)(65)a
How do we get the apothem?
72
72
How do we find ½ the central angle?
72
72
r
a
Central angle 360/5 72
(1/2) Central angle 36
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3
3
tan 36 3/a
a3/tan 36
Area(1/2)(65)3/tan 36
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One More Example
A (1/2)Pa
A (1/2)(86)a
(1/2) Vertex angle 30
60
60
60
60
60
a 4v3
30
A (1/2)(86) 4v3
A 96v3
4
4
8
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Area of Circles

• Formula from middle school,
• Area pr2
• Where Pi is the ratio of the Circumference/Diamete
  r and r is the radius of the circle.
• Find the exact area of a circle with a radius of
  5.
• Area pr2 p52 25p

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Circles and Regular Polygons
Regular Triangle inscribed in a circle
Area of Segments Circle Area Triangle Area
pr2 (1/2)Pa
r
a
s
Area of One Segment (1/3)(Circle Area ?
Area) (1/3)(pr2 (1/2)Pa)
Segments Area betweenchord and circle.
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11.4

• Areas of Irregular Figures

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Simple

• This is very simple if you know how to find the
  areas of standard figures.
• All you will need to do is see what is being
  added and or subtracted then write the formula.
• Example Area Big Circle Area of Triangle
  Area of Little Circle.

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11.5

• Probability

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Probability

• Four Cases
• Specific Examples
• Length Examples
• Area Examples
• Volume Example (Chap 13)
• Same formula
• Favorable Occurrence/Possible Occurrence

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Specific Example

• Put five marbles, two red and three blue into a
  bag. What is the probability of getting a blue
  one?
• Favorable/Possible
• 3 blue / 5 possible or 3/5 or .6 or 60
• What is the probability of not getting a blue
  one?
• If possible of getting a blue one is 60, then
  the probability of not getting a blue one is 40.

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Another example
What is the probability of picking two blues at
once?
What are the possible combinations?
_3_ 10
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Length Example
A
C
B
-3
11
2

• Find the probability of landing on segment AB if
  you can only land on segment AC.

• Prob AB/AC 5/14 or approx 36

• There are an infinite number of points on segment
  AB and segment AC so you need to use length

• Take the favorable length/possible length

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Area Example

• Find the probability of landing in the circle if
  you can only land in the square.

2
5
5

• There are an infinite number of points in the
  circle and the square so you have to use area.

• Prob Fav Area/Poss Area Acircle/Asquare

• Prob pr2/s2 p22/52 4p/25 50

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Sector Area

• Just like arc length is a portion of the
  circumference of the circle, sector area is a
  portion of the area of the circle.