area of parallelograms definition: a parallelogram is a quadrilateral with opposite sides parallel
TRANSCRIPT
Area Of ParallelogramsB
D C
A
Definition: A parallelogram is a quadrilateral with opposite sides parallel.
Review of Characteristics of Parallelograms
B
D C
A
Opposite sides are congruent.
Opposite angles are congruent.
Opposite sides are parallel. Consecutive angles are supplementary. (SSI angles)
Review of Characteristics of Parallelograms
Opposite sides are congruent.
Opposite angles are congruent.
Opposite sides are parallel. Consecutive angles are supplementary. (SSI angels)
B
D C
A
Diagonals bisect each other.
Areas of ParallelogramsFind the area by counting squares.
Full blocks =
Half blocks =
1( ) 22
4
Total Area =
2
5
84
10 su.
Areas of Parallelograms
Find the area by counting squares.
Full blocks =Half blocks =
1( ) 32
6
Total Area =
4
2
5
12
6
15 su.
Areas of Parallelograms
Find the area by counting squares.
Full blocks =
Half blocks =
1( ) 42
8
Total Area =
4
2
5 10
20
8
24 su.
It is tough to compute all the partial blocks.
There must be an easier way!
There is!!!
I like rectangles much better!They are real easy!
Don’t you agree?
4
2
5
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Doesn’t a parallelogram look
like a rectangle with it’s side kicked in?
Let’s cut off a corner and start to make a rectangle.
4
2
5
4
2
5
Wow! We have part of a rectangle.
Now watch what else we have.
4
2
5
Let’s move the triangle to the other side.
The two triangles are congruent.
4
2
5
4
2
5
We have now created a rectangle with the same area as the parallelogram.
What has happened?
4
2
5
We have now created a rectangle with the same area as the parallelogram.
Therefore, the formula for the area of a parallelogram is the same as that of a rectangle.
A bh
4
2
5
ParallelogramC
AD
B
b
h A bh
ParallelogramC
AD
B
b2 h2 A bh
Note that there is another base and another height. Sometimes you must use the other height.
Triangle
4
2
5
Now, we can use the parallelogram formula to derive the area of a triangle.
No, we are not going to add up squares again.
Triangle
2
5
Let’s construct a line through the vertex parallel to the base.
Triangle
2
5
Let’s construct another line through the right vertex parallel to another side.
We have just created a parallelogram.
Triangle4
2
5
We can do this for any triangle.
Note that both the triangles are congruent by…
SAS, SSS, ASA, or AAS
4
2
5
SAS
Opposite sides of a parallelogram are congruent.
Opposite sides of a parallelogram are congruent.
Opposite angles of a parallelogram are congruent.
4
2
5
SSS
Opposite sides of a parallelogram are congruent.
Opposite sides of a parallelogram are congruent.
Reflexive property.
4
2
5
ASA
Opposite sides of a parallelogram are congruent.
Opposite angles of a parallelogram are congruent.
AIA: Alternate Interior Angles are congruent.
4
2
5
AAS
Opposite sides of a parallelogram are congruent.
Opposite angles of a parallelogram are congruent.
AIA: Alternate Interior Angles are congruent.
TriangleAll triangles are just Half a parallelogram.
1
2A bh
Therefore…
TriangleNote where the height is located.
hIt is the height of both the parallelogram and triangle.
Sample Problems
14.75
8.25
Find the area.
A bh A
A= 121.6875
8.2514.75
A= 121.7
Finding Areas of Parallelogram
A bh A
12
20
16
12 16
192A
Find the area: fractions
354cm
1124cm
A bh A
A
354
1124
5 4 3
4
3 2354 4
12 4 11124 4
49
4
23
4
49
41,127
16A
77016
A
70.44A
Find the area: fractions
354cm
1124cm
A bh A
70.4375A3545.75
11
14
52 2.2
Convert to decimals.
5.7512.25
70.44A
Find the area: mixed modes
354
12.29
A bh A
Convert to decimals.
3545.75
5.7512.29
70.6675A
70.67A
Find the area: mixed modes
354
6 3
A bh
A
A = 59.754
Convert to decimals.
6 3 6 1.732 10.392
3545.75
10.392 5.75
A = 59.8
Find Area of Triangle
4
35
1
2A bh
1
2A
6A
4 3
Triangle ABC has three altitudes or heights.They are …
DB C
A
EF
, ,AD BE CF
Each side is a corresponding base.
Find the base associated with the corresponding height.
& _______CF
& ______BE
& ______AD BCAC
AB
If the area = 96 and ….
DB C
A
EF
12BCFind the values of AD and FC.
This is a backwards problem.You always start with the formula.
16AB
1
2A bh
1
2 1296 h
96 6h16 h
h
DB C
A
EF
If the area = 96 and ….
12BCFind the values of AD and FC.
This is a backwards problem.You always start with the formula.
16AB
1
2A bh
1
2 1696 h
96 8h12 h
h
Backward ProblemsIf the area is 125 sf, and the base is 25, find the height.
1
2A bh
1
2
25
25125 h
1125 25
2h2 2
150 25hDivide by 25
6 sf h
h
If the area is 182 sf, and the base is 30, find the height.
30
30
h
A bh h182
182 30h
182
30h
182
30h
With these problems, you…1 Plan solution with equations or written strategy as if you have all the information needed.
2. Write the equation, leaving empty parentheses to insert the needed values.
4. When all values are found, complete the original strategy or equation.
3. Go find the needed values in sidebar stages, substituting back into the original strategy or equation.
Compound Complex Multiple Stage Problems
Find Area of Triangle
13
12
51
2A bh
1
2A
5, 12, 13 triangle
12 5
30A
Find Area of Triangle
8
88
600
1
2A bh
1
2A
16 3A30
1
2
60
84
2sf
4 3
8 4 34
27.712A27.7A
Find Area of Triangle
8
60015
120
1
2A bh
1
2A 15
30-60-90 triangle
30
1
2
6084
2sf
4 3
4 3
30 3A51.96A52A
Find Area of Triangle
11
b
5
1
2A bh
1
2A 2 2 2 5 11
b5
225 121b 2 96b
9 796 . 97b
9.8
24.5A
Find Area of Parallelogram
60
A bh A
8
1616
30
1
2
60
h
Need to find value of height.
84
2sf
4 34 3
110.851A su64 3A
110.9 uA s
Find Area of Triangle
815
1
2A bh
458 2
1
2A
45-45-90 triangle
1
145
15
8 28
2sf
8
60A
Find the area of pentagon.
10
10
10
1010Add line and label figure.
10 rect triangleA A A
triA bh bh 1
2A 101010
600
30
1
2
60 0
25
1sf
5 3
5 3
100 25 3A
5
100 43.301A 143.301A
C
A B
EDC
A B
ED
C
A B
ED
C
A B
ED
Which triangle has the largest area?
ABC or ABD or ABE They all have the same area.
Why?
C
A B
ED
C
A B
ED
C
A B
ED
1
2A bh
They each have the same base: AB
They each have the same height.
MB C
AM is midpoint of BC
h h is 13 cm.
24mBC cm
Find the area of triangles ABM and ACM.
12BM MC
h is 13 for both triangles.
1
2A bh
112 13
2A
12 12
MB C
AM is midpoint of BC
h h is 13 cm.
24mBC cm
1
2A bh
112 13
2A
278A cm
12 12
Summary1. The area of a parallelogram is…
2. The area of a triangle is half the area of parallelogram.
3. A triangle has three heights or altitudes.
4. A triangle has three bases (sides) to correspond with each height.
A bh
1
2A bh