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AREA FORMULAS © 2009 AIMS Education Foundation
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ISBN 978-1-60519-009-9
Printed in the United States of America
Area Formulas for
Parallelograms, Triangles, and
Trapezoids
Area Formulas for
Parallelograms, Triangles, and
Trapezoids
AREA FORMULAS 5 © 2009 AIMS Education Foundation
Table Table of ContentsTable Table ofof Contents Contents
Area Formulas Area Formulas forfor Parallelograms, Parallelograms, Triangles, and Trapezoids
Area Formulas Area Formulas forfor Parallelograms, Parallelograms, Triangles, Triangles, andand Trapezoids Trapezoids
The area of a parallelogram is found by multiplying base by height. All
parallelograms that have the same length base and height have the same
area no matter how far they have been skewed.
Lesson One: Parallelogram Cut-Ups ........................................... 9
Parallelogram Cut-Ups ............................................................. 10
By measuring the base, height, and sides of parallelograms, students recognize that the height and sides of parallelograms are different lengths. They learn to use the appropriate measures to determine the perimeters of parallelograms.
By cutting up a parallelogram and reforming it into a rectangle, students discover the relationship of the two and the similarity of fi nding the areas of both. A critical understanding is being able to differentiate between a side and the height. Students should come to recognize that any of the sides can be designated as the base.
Parallelogram Cut-Ups .............................................................................. 15
The comic summarizes the relationship of a parallelogram to a rectangle with the same base and height and develops the meaning of the general formula for area.
Lesson Two: Areas on Board: Parallelograms .................. 17
Areas on Board: Parallelograms .............................................. 19
By recognizing that every parallelogram can be transformed into an equal area rectangle, students confi rm the area formula of base times height.
DayDay33
InvestigationInvestigation
DaysDays1 1 andand 2 2
InvestigationInvestigation
ComicComic
BIG IDEA:
Welcome to the AIMS Essential Math Series! .................... 3
AREA FORMULAS 6 © 2009 AIMS Education Foundation
A Shifting Parallelogram ................................................................ 21
A parallelogram is skewed while keeping its sides the same length resulting in a changed height and area. The parallelogram is then skewed again, this time changing the length of the sides to keep the height constant resulting in a constant area. The visual display accentuates the critical nature of the height.
Parallelograms ..................................................................23
These problems provide an assessment of understanding of the area formula as students are asked to fi nd various unknown dimensions.
A triangle is half the area of a parallelogram with the same height and width.
The formula A = (b • h)/2 = ½ (b • h) describes this relationship.
Lesson Three: Triangle Cut-Ups ....................................................25
Triangle Cut-Ups ......................................................................26
By matching pairs of congruent triangles and forming parallelograms with them, students will recognize that a triangle is half of a parallelogram. This understanding connects with the formula for the area of a triangle as: A = (b • h)/2 = ½ (b • h).
Triangle Cut-Ups .......................................................................................29
The comic emphasizes the relationship of two congruent triangles to a parallelogram and develops the formula by showing how it represents this relationship.
Lesson Four: Triangles to Parallelograms ............................. 31
Triangles to Parallelograms ......................................................32
All triangles can be cut so their pieces can be reformed into parallelograms. A parallelogram will have a base or height that is half the base or height of the triangle from which it was made. The experience provides a visual model of the formula in the form A = ½ b • h = b • ½ h.
Triangles to Parallelograms .......................................................................35
The comic emphasizes the relationship of two congruent triangles to a parallelogram and develops the formula by showing how it represents this relationship.
DayDay77
ComicComic
DayDay55
Investigation Investigation
Problem SolvingProblem Solving
InvestigationInvestigation
BIG IDEA:
DayDay66
ComicComic
AnimationAnimationDayDay44
AREA FORMULAS 7 © 2009 AIMS Education Foundation
Lesson Five: Areas on Board: Triangles .................................. 37
Areas on Board: Triangles ........................................................39
A geoboard or dot paper provides a grid for counting square area. By looking at triangles with equal areas, students fi nd that the triangles have a common base and height. Multiplying the base and height gives the area of a rectangle that is twice the size of the triangle.
Triangle Transformations ............................................................... 41
A triangle is transformed into a parallelogram or rectangle in four ways. The vivid visual models reinforce the understanding of the area formula and demonstrate several forms of the formula.
Practice: Bigger Triangles .........................................................................................43
Using larger triangles drawn on dot paper, students apply and practice what they learned from the initial investigation.
Triangles ...........................................................................45
Students use the area formula to solve problems with triangles.
The area of a trapezoid is a calculated by multiplying the average base
by the height. The formula is
A = ((b1 + b2)/2) • h = ((b1 + b2) • h) / 2 = ½ ((b1 + b2) • h).
Lesson Six: Trapezoid Cut-Ups .......................................................49
Trapezoid Cut-Ups...................................................................50
Cutting out and combining two trapezoids into a parallelogram demonstrates the area formula Any two congruent trapezoids form a parallelogram that has a base that is the combined lengths of the top and bottom bases of the trapezoid. Dividing the area of the parallelogram by two gives the area of the trapezoid.
Trapezoid Cut-Ups ...................................................................................53
The comic demonstrates that two congruent trapezoids form a parallelogram and reinforces how the formula represents this relationship.
DayDay1111
DayDay1010
Problem SolvingProblem Solving
InvestigationInvestigation
BIG IDEA:
A =(b1 + b2) • h
2
DayDay99
AnimationAnimation
.
InvestigationInvestigationDayDay88
ComicComic
AREA FORMULAS 8 © 2009 AIMS Education Foundation
Lesson Seven: Trapezoids to Parallelograms .....................55
Trapezoids to Parallelograms ..................................................56
Cutting a trapezoid in half and rotating it forms a parallelogram of the same area. Calculating the area of the parallelogram, which is half the height of the trapezoid, gives the area of the trapezoid. The transformation of the trapezoid is a visual model of the formula in the form A = ½h • (b1 + b2).
Trapezoid Tumbles ........................................................................ 57
The animation demonstrates both doubling the trapezoid to make a parallelogram twice as big as the trapezoid and cutting the trapezoid in half to make a parallelogram equal in size. The dynamic visual reinforces the ideas developed in the investigations and encourages students to seek the commonalities of the formulas to clarify the concepts expressed in the formulas.
Trapezoids ........................................................................59
Students apply and practice using a visual model or an area formula to solve trapezoid problems.
Polygon Puzzle ................................................................. 61
A fi ve-piece puzzle can be formed into two rectangles, two parallelograms, one triangle, and three trapezoids. It provides an opportunity to review and summarize the meaning of the formulas and see their interrelatedness.
Geoboard Designs ......................................................................65
Some very interesting and complex shapes are made by combining polygons on dot paper. The problems can be solved visually or by using formulas.
Glossary ............................................................................................................................................69National Standards and Materials .................................................................................................... 71Using Comics to Teach Math ............................................................................................................72Using Animations to Teach Math .....................................................................................................73The Story of Area Formulas ..............................................................................................................75The AIMS Model of Learning ............................................................................................................79
DayDay1515
Problem SolvingProblem SolvingDayDay1414
Problem SolvingProblem Solving
AssessmentAssessment
DayDay1313
AnimationAnimation
InvestigationInvestigationDayDay1212
AREA FORMULAS 9 © 2009 AIMS Education Foundation
1. What two dimensions did you use to determine the perimeter?
2. How did you use those dimensions to fi nd the perimeter?
3. What two dimensions do you use to determine the area of the rectangle made from the two pieces of the parallelogram?
4. How are the dimensions used for area different than the ones used for perimeter?
5. Does it matter which side is the base?
6. Write a formula to describe how you use these two dimensions to get the area (A) of the parallelogram.
ARALLELOGRAMP UTC PSUthese
are some pressing questions.
the lengths of each pair of opposite sides
fi nd the sum of all four sides
base and height
Base is a side, but height is at right angles to the base. For the
rectangle, the base and height are the same as the sides.
No, as long as you measure the height perpendicular from the base
you chose.
Area = base • height, A = b • h
AREA FORMULAS 13 © 2009 AIMS Education Foundation
A
B
CD
A
B
C
D
find the perimeterby adding up the
four sides.
measure thebase...
...and the height.
cut and arrangethe parallelograms
into rectangles.
ParallelogramA
ShortSide (cm)
LongSide (cm)
Perimeter(cm)
Long Base
Short Base
ParallelogramB
Long Base
Short Base
ParallelogramC
Long Base
Short Base
Base(cm)
Height(cm)
Area(cm2)
ARALLELOGRAMP UTC PSUHow is fi nding the area of a parallelogram different from fi nding the area of a rectangle?
Use the Measuring Pad to fi nd and record the lengths of the sides of each parallelogram. (Round to the nearest whole centimeter, if necessary.)
Determine and record the perimeter of each parallelogram.
Make the long side of each parallelogram the base. Measure and record the length of the base and the height.
Make the short side of each parallelogram the base. Measure and record the length of the base and the height.
Cut a dotted line marking one of the heights of the parallelogram.
Cut different lines on each pair of congruent parallelograms.
Make a rectangle using the two pieces from each parallelogram.
Determine and record the area of each parallelogram by fi nding the area of its two pieces making the rectangle.
1.
2.
3.
1. 2. 3.
8
9
12
12
12
16
40
42
56
12
12
16
8
9
12
8
6
9
12
8
12
96
72
144
96
72
144
How is fi nding the area of a parallelogram different from fi nding the area of a rectangle?
It is crucial to be able to differentiate among a side, the base, and the height. A parallelogram has four sides. The base is one of these sides. The height is only a side of a parallelogram when the parallelogram is a rectangle. Using the Measuring Pad, students focus on these differences. By cutting up a parallelogram and reforming it into a rectangle, one discovers the relationship of the two and the similarity of fi nding area of both.
MaterialsScissorsParallelogramsMeasuring Pad
Investigation
Investigation
Investigation
Investigation
Investigation
Investigationtigtigtigtigtigtig
ARALLELOGRAMP UTC PSU
Area is the
product of base and height. See
that students recognize the
difference between
height and side.
Make sure students align each side to the ruler as they
measure.
Confirm that students
recognize that the areas of the
parallelogram and the corresponding
rectangle are equal.
Each group of 4 or 5
students will need 2 of each of the
parallelograms.
Compare the two sides of the chart so it is
recognized that any side can be used as the base but the height is not a side except
in the rectangle.
Students reinforce their understanding of the relationship of a parallelogram to a rectangle when fi nding perimeter and area.Comics
ComicsComicsComicsComicsComics
AREA FORMULAS 12 © 2009 AIMS Education Foundation
AREA FORMULAS 10 © 2009 AIMS Education Foundation
your groupwill need two
of eachparallelogram.
cut out the parallelogramsalong the bold
lines.
C
A
B
ARALLELOGRAMPUTC PSU
AREA FORMULAS 11 © 2009 AIMS Education Foundation
0
10
23
4
1 2 3 4 5 6 7 8 9
10
11
12
13
14
15
16
56
78
91
01
11
21
31
41
51
61
71
81
92
02
12
2
BA
SE
HEIGHT
centim
eters
centimetersM
easurin
g P
ad
AREA FORMULAS 12 © 2009 AIMS Education Foundation
A
B
CD
A
B
C
D
find the perimeterby adding up the
four sides.
measure thebase...
...and the height.
cut and arrangethe parallelograms
into rectangles.
ParallelogramA
ShortSide (cm)
LongSide (cm)
Perimeter(cm)
Long Base
Short Base
ParallelogramB
Long Base
Short Base
ParallelogramC
Long Base
Short Base
Base(cm)
Height(cm)
Area(cm2)
ARALLELOGRAMP UTC PSUHow is fi nding the area of a parallelogram different from fi nding the area of a rectangle?
Use the Measuring Pad to fi nd and record the lengths of the sides of each parallelogram. (Round to the nearest whole centimeter, if necessary.)
Determine and record the perimeter of each parallelogram.
Make the long side of each parallelogram the base. Measure and record the length of the base and the height.
Make the short side of each parallelogram the base. Measure and record the length of the base and the height.
Cut a dotted line marking one of the heights of the parallelogram.
Cut different lines on each pair of congruent parallelograms.
Make a rectangle using the two pieces from each parallelogram.
Determine and record the area of each parallelogram by fi nding the area of its two pieces making the rectangle.
1.
2.
3.
1. 2. 3.
AREA FORMULAS 13 © 2009 AIMS Education Foundation
1. What two dimensions did you use to determine the perimeter?
2. How did you use those dimensions to fi nd the perimeter?
3. What two dimensions do you use to determine the area of the rectangle made from the two pieces of the parallelogram?
4. How are the dimensions used for area different than the ones used for perimeter?
5. Does it matter which side is the base? Explain.
6. Write a formula to describe how you use these two dimensions to get the area (A) of the parallelogram.
ARALLELOGRAMP UTC PSUthese
are some pressing questions.
AREA FORMULAS 15 © 2009 AIMS Education Foundation
Para
llelo
gra
m C
ut-U
ps
THIN
GS T
O LO
OK F
OR:
AR
EA
OF
PA
RA
LL
EL
OG
RA
MS
, T
RIA
NG
LE
S, A
ND
TR
AP
EZ
OID
SE
SS
EN
TIA
L M
AT
H S
ER
IES
1
1. W
hat
do
yo
u k
no
w a
bo
ut
the s
ides
of a
parall
elo
gram
?
2.
Ho
w d
o y
ou f
ind t
he p
erim
ete
r
o
f a
parall
elo
gram
?
5.
Wha
t is
the
fo
rm
ula
fo
r f
indin
g t
he a
rea o
f a
parall
elo
gram
?
3.
Wha
t is
the
base o
f a
parall
elo
gram
?
4. H
ow d
o y
ou f
ind t
he h
eig
ht
o
f a
parall
elo
gram
?
Like f
or
parallelo
gram
C,
it w
as 1
2 p
lus 1
6plus 1
2 p
lus 1
6.
that’s
56
.
The m
easurin
g p
ad w
as l
ike t
wo
rulers
that
were p
erpendic
ular
to
each o
ther.
It w
as
easy
to
put o
ne s
ide o
f a p
arallelo
gram
on
one o
f the r
ulers a
nd m
eas
ure
it.
I j
ust
multip
lie
d t
he
lo
ng
sid
e a
nd t
he
sho
rt s
ide e
ach
by
2 a
nd t
hen
added t
hem
!
Okay,
ho
w m
any
sid
es d
id y
ou h
ave
to
measure?
That’s
rig
ht,
mark.
For e
very
parallelo
gram
, the
sid
es a
cro
ss f
ro
m e
ach
other a
re n
ot o
nly
parallel
, but t
hey
are a
lso
eq
ual
in l
eng
th.
We c
all
them
oppo
sit
es
ides
. So
, w
e c
an
say
that o
ppo
sit
esid
es o
f a
parallelo
gram
have
eq
ual
leng
th.
What e
ls
edid
yo
u d
o?
Aft
er m
easurin
g,
we a
dded u
p t
he l
eng
ths
of
the f
our s
ides t
o f
ind
the p
erim
eter
.
The p
erim
eter
is h
ow
far
it is
aro
und t
he
parallelo
gram
.
It w
as
pretty
far.
Aft
er y
ou c
ut o
ut t
he
parallelo
gram
s a
tthe b
eg
innin
g o
f this
activ
ity,
yo
u m
easured
so
me l
eng
ths.
Let’s
talk a
bo
ut t
hat.
Yeah, w
e m
easured
the s
ides.
We u
sed
this
pad
thin
g t
om
easure t
hem
.
We o
nly
needed t
om
easure t
wo
sid
es
because t
he s
ides
that a
re a
cro
ss
fro
m e
ach o
ther
are t
he s
am
eleng
th.
01
02
34
12345678910111213141516
56
78
910
1112
1314
1516
1718
1920
2122
BASE
cent
imet
ers
centimeters HEIGHTC
C
12
+ 1
6 +
12
+ 1
6 =
56
12
16
Mea
suring
Pad
2
Well d
one.
Eit
her w
ay
is f
ine
fo
r f
indin
gthe p
erim
eter.
The n
ext t
hin
gw
e d
id w
as t
o f
ind
the a
rea o
f each o
fthe p
arallelo
gram
s,
rig
ht?
Well, w
e f
irst
had t
o m
easure
the h
eig
hts
of
the
parallelo
gram
s.
For e
ach
bas
e t
here is
a h
eig
ht t
hat
go
es w
ith it.
Ho
ld o
n a
min
ute, va
nessa,
that is a
very
go
od
observa
tio
n.
Wait
, no
w t
he
heig
ht c
hang
ed
.It’s
no
t 1
2anym
ore?
Well, ye
ah,
red, that’s
pretty
obvi
ous.
Yeah, Red,
thin
gs c
hang
ew
hen y
ou u
se
a d
iffe
rent s
ide
for t
he b
ase.
The p
erim
eter f
or
parallelo
gram
Cw
as 2
tim
es 1
2plus 2
tim
es 1
6.
usin
g t
he p
ad, w
e r
ested o
ne
of t
he s
ides o
f t
he
parallelo
gram
on t
he b
otto
m r
uler a
nd m
easured
the h
eig
ht o
f t
he p
erpendic
ular d
otted l
ine.
Like if
parallelo
gram
C is r
estin
g o
n t
he
lo
ng
sid
e, then t
hat’s
the b
ase a
nd t
hat’s
16
. If
you m
easure t
he h
eig
ht f
ro
m t
hat b
ase it’s
9.
So
, w
hen
parallelo
gram
C is
restin
g o
n t
he s
ho
rt
sid
e, the h
eig
ht is
12
. N
ow
we k
no
who
w t
all it is!
I g
et it, the
parallelo
gram
is
taller
when it’s
restin
g o
n t
he s
ho
rt
sid
e, and it’s
sho
rter
when it’s
restin
g o
nthe l
ong
sid
e.
Yeah, and w
hateve
r s
ide
the p
arallelo
gram
is r
es
tin
g o
n, w
ecall t
hat t
he
bas
e.
01
02
34
12345678910111213141516
56
78
910
1112
1314
1516
1718
1920
2122
BASE
cent
imet
ers
centimeters HEIGHT
C
01
02
34
12345678910111213141516
56
78
910
1112
1314
1516
1718
1920
2122
BASE
cent
imet
ers
centimeters HEIGHT
C
C
2 •
12
+ 2
• 1
6 =
56
12
16
AREA FORMULAS 16 © 2009 AIMS Education Foundation
3
Do
es t
hat
mean w
e h
ave
to
kno
w t
wo
dif
ferent
form
ulas f
or t
he a
rea
of
a r
ectang
le?
And I
wanted
to
sho
w y
ou h
ow
that h
elped u
s f
ind
a f
orm
ula f
or f
indin
gthe a
rea o
f any
parallelo
gram
.
It’s
because
we a
lready
knew
the f
orm
ula t
o f
ind
the a
rea o
f a
rectang
le
.
So
, class, ho
wdid
we d
o it? H
ow
did
we f
igure o
ut t
he
fo
rm
ula f
or f
indin
gthe a
rea o
f any
parallelo
gram
?
But r
ight n
ow
,w
e c
an u
se
base
tim
es h
eig
ht t
o h
elp
us f
igure o
ut a
form
ula f
or t
he
area o
f any
parallelo
gram
.
We j
ust c
ut u
pthe p
arallelo
gram
and p
ut it b
ack
to
gether t
o m
ake
a r
ectang
le
.It w
as f
un!
And t
he
parallelo
gram
and t
he r
ectang
le
have
the s
am
e a
rea
because t
hey’
re b
oth
made o
ut o
f the
sam
e p
ieces.
We f
ound
out t
hat a
no
ther
way
to
thin
k a
bo
ut
the f
orm
ula f
or a
rea
of
a r
ectang
le is
that it’s
base
tim
es h
eig
ht.
That’s
exactly
rig
ht,
Mark.
What w
e f
ound o
ut w
as t
hat l
eng
th a
nd w
idth
on a
rectang
le a
re t
he s
am
e t
hin
g a
s b
ase
and h
eig
ht.
So
, a r
ectang
le h
as a
base a
nd
heig
ht j
ust l
ike a
ll t
he r
est o
f the
parallelo
gram
s.
If
12
is t
he b
ase o
f the r
ectang
le, then
the h
eig
ht is 8
, rig
ht? A
nd t
he a
rea is b
ase t
imes
heig
ht, that’s
12
tim
es 8
or 9
6.
Class, there is a
nim
po
rtant r
easo
nw
hy
we included t
he
rectang
le a
lo
ng
wit
h t
he o
ther t
wo
parallelo
gram
s.
No
t r
eally,
redm
ond.
You’l
l u
sually
thin
kabo
ut t
he a
rea o
f a
rectang
le a
s l
eng
th
tim
es w
idth.
Yeah, w
e a
lready
knew
that t
he f
orm
ula f
or
findin
g t
he a
rea o
f a
rectang
le is l
eng
th
tim
es
wid
th.
01
02
34
12345678910111213141516
56
78
910
1112
1314
1516
1718
1920
2122
BASE
cent
imet
ers
centimeters HEIGHT
01
02
34
12345678910111213141516
56
78
910
1112
1314
1516
1718
1920
2122
BASE
cent
imet
ers
centimeters HEIGHT
A
8
12
A
A
4
Is t
hat t
he
form
ula f
or t
he a
rea
of
a p
arallelo
gram
?Is it b
ase t
imes h
eig
ht
for e
very
parallelo
gram
?
And y
ou c
an
use t
he s
ho
rt s
ide
for t
he b
ase o
r y
ou
can u
se t
he l
ong
sid
e, rig
ht?
That is a
n e
xcellent
sum
mary,
redm
ond.
You’v
e g
ot it!
That’s
rig
ht,
juana.
You j
ust h
ave
to
be s
ure t
hat y
ou
measure t
he h
eig
ht
fro
m t
hat b
ase.
But f
or a
parallelo
gram
the
area is j
us
t b
ase
tim
es h
eig
ht.
I t
hin
k I
’ve
go
t it!
The a
rea o
fthe r
ectang
le is l
eng
th
tim
es w
idth o
r b
ase
tim
es h
eig
ht!
They
mean
the s
am
e t
hin
g,
rig
ht?
So
, if
base t
imes
heig
ht t
ells u
s t
he
area o
f the r
ectang
le
,then b
ase t
imes h
eig
ht
tells u
s t
he a
rea o
fthe p
arallelo
gram
as w
ell!
Like f
or p
arallelo
gram
C, if
we c
ut it u
p into
tw
o p
ieces, w
e c
an m
ove
that t
ria
ng
le t
o t
he
other s
ide
and it m
akes a
rectang
le.
You c
an t
urn t
he p
arallelo
gram
into
arectang
le a
nd b
oth s
hapes h
ave
the s
am
e b
ase
and h
eig
ht.
They
also
have
the s
am
e a
rea.
The s
am
e t
hin
ghappens w
hen y
ou c
ut
up p
arallelo
gram
B.
The a
rea o
f the
rectang
le is b
ase
tim
es h
eig
ht, o
r 9
tim
es 1
6, sam
e a
s t
he
parallelo
gram
.
So
, if
the a
rea o
f the
rectang
le is b
ase
tim
es h
eig
ht, then
the a
rea o
f the
parallelo
gram
is
base t
imes h
eig
ht
as
well
.
It is,
redm
ond.
For e
very
parallelo
gram
, if
yo
um
easure o
ne o
f the s
ides, that’s
called t
he b
as
e.
And if
you t
hen
measure t
he h
eig
ht f
ro
m t
hat
base, then t
he a
rea is b
ase
tim
es h
eig
ht.
C
9
16
C
9
16
C
9
16
C
9
16
B
8
9
B
8
9
B
8
9
h
b