Area Formulas Parallelograms, Triangles, Trapezoids Area

AREA FORMULAS © 2009 AIMS Education Foundation

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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


ComicComic

BIG IDEA:

Welcome to the AIMS Essential Math Series! .................... 3


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.

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Investigation Investigation

Problem SolvingProblem Solving


BIG IDEA:

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AnimationAnimationDayDay44


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.

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BIG IDEA:

A =(b1 + b2) • h

2

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AnimationAnimation

.

InvestigationInvestigationDayDay88

ComicComic


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

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Problem SolvingProblem SolvingDayDay1414


AssessmentAssessment

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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


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



your groupwill need two

of eachparallelogram.

cut out the parallelogramsalong the bold

lines.

C

A

B

ARALLELOGRAMPUTC PSU


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


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.


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.

KEEP GOINGKEEP GOING


Para

llelo

gra

m C

ut-U

ps

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GS T

O LO

OK F

OR:

AR

EA

OF

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RA

MS

, T

RIA

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LE

S, A

ND

TR

AP

EZ

OID

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SS

EN

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L M

AT

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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.

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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?

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rig

ht,

mark.

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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.

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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.

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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

.

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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

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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

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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


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

Documents

Area Formulas Parallelograms, Triangles, Trapezoids Area