Transcript Area Formulas - Kendriya Vidyalaya Malleswaram, Bangalore

Area Of Parallelograms
A
D
B
C
Definition:
A parallelogram is a quadrilateral
with opposite sides parallel.
Review of Characteristics
of Parallelograms
A
D
Opposite sides are parallel.
Opposite sides are congruent.
Opposite angles are congruent.
B
C
Consecutive angles are
supplementary. (SSI angles)
Review of Characteristics
of Parallelograms
B
A
D
Opposite sides are parallel.
C
Opposite sides are congruent.
Consecutive angles are
supplementary. (SSI angels)
Opposite angles are congruent.
Diagonals bisect each other.
Areas of Parallelograms
Find the area by counting squares.
2
5
8
-2Half blocks = 4
Full blocks =
10
1
4( )  2
2
Total Area = 10 su.
Areas of Parallelograms
4
2
Find the area by counting squares.
5
-2
-4
Full blocks = 12
Half blocks = 6
10
15
1
6( )  3
2
Total Area = 15 su.
6
Areas of Parallelograms
4
Find the area by counting squares.
2
Full blocks = 20
5
-2
-4
Half blocks = 8
10
15
1
8( )  4
2
Total Area = 24 su.
-6
4
2
?
?
?
?
?
?
?
?
5
-2
-4
It is tough to compute
all the partial blocks.
There must be
an easier way!
There is!!!
10
I like rectangles much better!
They are real easy!
Don’t you agree?
4
2
5
-2
-4
-6
10
15
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
-2
10
Wow! We have part of a rectangle.
Now watch what else we have.
-4
15
4
2
5
-2
10
15
The two triangles are congruent.
Let’s move the triangle to the other side.
-4
4
10
What has happened?
2
8
6
5
-2
4
10
15
10
15
We have now created
a rectangle
with the same area
as the parallelogram.
-4
2
-6
5
10
4
We have now created
a rectangle
with the same area
as the parallelogram.
8
2
6
5
4
-2
2
-4
-6
5
Therefore,
10
15
the formula for the area
of a parallelogram
is the same
as that of a rectangle.
A  bh
10
15
Parallelogram
C
B
A  bh
h
A
b
D
Parallelogram
C
B
b2
A
h2
A  bh
D
Note that there is another base and another height.
Sometimes you must use the other height.
Triangle
No, we are not going
to add up squares again.
6
4
Now, we can use the
parallelogram formula
to derive
the area of a triangle.
2
5
10
6
Triangle Let’s construct a line through
4
the vertex parallel to the base.
2
5
-2
10
6
Triangle
Let’s construct another line through
the right vertex parallel to another side.
4
2
5
-2
10
We have just created a parallelogram.
6
Triangle
We can do this for any triangle.
4
2
5
10
Note that both the triangles are congruent by…
-2
SAS, SSS, ASA, or AAS
Opposite sides of a parallelogram are congruent.
6
Opposite angles of a parallelogram are congruent.
Opposite sides of a parallelogram are congruent.
4
2
SAS
5
10
Opposite sides of a parallelogram are congruent.
6
Opposite sides of a parallelogram are congruent.
Reflexive property.
4
SSS
2
5
10
AIA: Alternate Interior Angles are congruent.
6Opposite sides of a parallelogram are congruent.
Opposite angles of a parallelogram are congruent.
4
2
ASA
5
10
AIA: Alternate Interior Angles are congruent.
6
Opposite angles of a parallelogram are congruent.
Opposite sides of a parallelogram are congruent.
4
2
AAS
5
10
Triangle
All triangles are just
Half a parallelogram.
Therefore…
1
A  bh
2
Triangle
Note where the
height is located.
h
It is the height of both
the parallelogram
and triangle.
Sample Problems
Find the area.
8.25
14.75
A  bh
A 14.75  8.25 
A= 121.6875
A= 121.7
Finding Areas of Parallelogram
12
16
20
A  bh
A 
12
 16 
A 192
Find the area: fractions
A  bh
3
5 cm
4
1
12 cm
4
1 12  4   1  49
12  

4 
4
 4
3  5  4   3  23
5 

4  4  4
A 
A 
1
12
4
49
4


3
5
4
23
4


1,127
A
16
7
A  70
16
A 70.44
Find the area: fractions
3
5 cm
4
1
12 cm
4
Convert to decimals.
A  bh
A 12.25 5.75 
1
12  12.25
4
A 70.4375
3
5  5.75
4
A 70.44
Find the area: mixed modes
3
5
4
12.29
Convert to decimals.
3
5  5.75
4
A  bh
A 12.29 5.75
A 70.6675
A 70.67
Find the area: mixed modes
3
5
4
6 3
A  bh
A 10.3925.75 
Convert to decimals.
6 3  6 1.732 10.392
3
5  5.75
4
A = 59.754
A = 59.8
3
Find Area of Triangle
1
A  bh
5
2
1
A   4  3 
4
2
A 6
A
Triangle ABC has three altitudes or heights.
They are …
AD, BE, CF
Each side is a corresponding base.
F
B
E
D
C
Find the base associated with
the corresponding height.
AD & ______
BC
AC
BE & ______
CF & _______
AB
A
If the area = 96 and ….
BC 12
h
F
B

D
1
A  bh
2
AB 16
Find the values of AD and
E
C
FC.
This is a backwards problem.
You always start
with the formula.
1
96   12  h
2

96  6h
16  h
A
If the area = 96 and ….
BC 12
F
B

h
D
1
A  bh
2
AB 16
Find the values of AD and FC.
E
C
This is a backwards problem.
You always start
with the formula.
1
96   16  h
2

96  8h
12  h
Backward Problems
If the area is 125 sf,
and the base is 25, find the height.
h
2
25
150  25h
1
A  bh
2
1
125   25  h
2
1
125  25h 2
2
Divide by 25

6 sf  h
h
30
A  bh
182  30  h 
182  30h
If the area is 182 sf,
and the base is 30,
find the height.
182
h
30
182
h
30
Compound Complex
Multiple Stage Problems
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.
3. Go find the needed values in sidebar stages,
substituting back into the original strategy or equation.
4. When all values are found,
complete the original strategy or equation.
Find Area of Triangle
5
13
12
5, 12, 13 triangle
1
A  bh
2
1
A  12  5
2
A 30

Find Area of Triangle
8
8
1
A  bh
4 3
60
2
1 4
8
8
A   8 4 3 
sf   4
2
2
A16 3
0
A 27.712
A 27.7
Find Area of Triangle
1
A  bh
8 4 3
2
120
15
30-60-90 triangle
8
sf   4
2
600
1
A  15  4 3
2
A 30 3
A 51.96
A 52
Find Area of Triangle
1
11
A  bh
5
2
1
b
2
2
2
A



9.8
5
 5    b    11
2
25  b  121
2
b  96
2
b  96  9.797
A 24.5

Find Area of Parallelogram
8
60
h4 3
16
A  bh
A  16 4 3 
Need to find value of height.
8
sf   4
2
A 64 3
A 110.851 su
A 110.9 su
Find Area of Triangle
1
A  bh
8 2
8
2
45
15
45-45-90 triangle
1
A  15  8
2
A 60
8 2
sf 
8
2

Find the area of pentagon.
10
600
10
Add line and label figure.
5 3
A  Arect  Atriangle
10
10
10
10
A  bh  bhtri
1 5
A   1010    105 3
10
sf   5
2
2
A100  25 3
A100  43.301
A143.301
C
D
A
C
E
B
C
D
A
B
ABD or
B
C
A
E
Which triangle has
the largest area?
E
ABC or
A
D
D
E
B
They all have
the same area.
Why?
ABE
C
D
A
1
A  bh
2
B
C
D
A
E
D
E
B
They each have
the same base: AB
They each have
the same height.
B
C
A
E
A
M is midpoint of
BC
h is 13 cm.
h
B
12
M
mBC  24cm
12 C
Find the area of triangles ABM and ACM.
BM  MC 12
h is 13 for both triangles.
1
A  bh
2
1
A  12 13
2
A
M is midpoint of
BC
h is 13 cm.
h
B
12
M
mBC  24 cm
12 C
1
A  bh
2
1
A  12 13
2
A  78 cm
2
Summary
A  bh
1. The area of a parallelogram is…
2. The area of a triangle is half the area of parallelogram.
1
A  bh
2
3. A triangle has three heights or altitudes.
4. A triangle has three bases (sides)
to correspond with each height.