Area Formulas
for
Parallelograms, Triangles,
and
Trapezoids
Developed and Published
by
AIMS Education Foundation
This book contains materials developed by the AIMS Education Foundation. AIMS (Activities Integrating
Mathematics and Science) began in 1981 with a grant from the National Science Foundation. The non-profit
AIMS Education Foundation publishes hands-on instructional materials that build conceptual understanding.
The foundation also sponsors a national program of professional development through which educators may
gain expertise in teaching math and science.
Copyright © 2009 by the AIMS Education Foundation
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ISBN 978-1-60519-009-9
Printed in the United States of America
AREA FORMULAS
© 2009 AIMS Education Foundation
Area Formulas for Parallelograms,
Triangles, and Trapezoids
Table of Contents
Welcome to the AIMS Essential Math Series! .................... 3
BIG IDEA:
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
Days
1 and 2
Investigation 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
ﬁnding 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.
Comic 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
Day
3
AREA FORMULAS
Investigation Areas on Board: Parallelograms .............................................. 19
By recognizing that every parallelogram can be transformed into
an equal area rectangle, students conﬁrm the area formula of base
times height.
5
© 2009 AIMS Education Foundation
Day
4
Day
5
Animation 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.
Problem Solving Parallelograms ..................................................................23
These problems provide an assessment of understanding of the area
formula as students are asked to ﬁnd various unknown dimensions.
BIG IDEA:
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
Day
6
Investigation 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).
Comic 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
Day
7
Investigation 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.
Comic 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.
AREA FORMULAS
6
© 2009 AIMS Education Foundation
Lesson Five: Areas on Board: Triangles .................................. 37
Day
8
Day
9
Investigation 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 ﬁnd 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.
Animation 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.
Day
10
Problem Solving Triangles ...........................................................................45
Students use the area formula to solve problems with triangles.
BIG IDEA:
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
Day
11
Investigation Trapezoid Cut-Ups...................................................................50
Cutting out and combining two trapezoids
(b1 + b2) • h
.
into a parallelogram demonstrates the area formula A =
2
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.
Comic Trapezoid Cut-Ups ...................................................................................53
The comic demonstrates that two congruent trapezoids form
a parallelogram and reinforces how the formula represents this
relationship.
AREA FORMULAS
7
© 2009 AIMS Education Foundation
Lesson Seven: Trapezoids to Parallelograms .....................55
Day
12
Day
13
Investigation 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).
Animation 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.
Problem Solving Trapezoids ........................................................................59
Students apply and practice using a visual model or an area formula
to solve trapezoid problems.
Day
14
Day
15
Problem Solving Polygon Puzzle ................................................................. 61
A ﬁve-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.
Assessment 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 ............................................................................................................................................69
National Standards and Materials.................................................................................................... 71
Using Comics to Teach Math ............................................................................................................72
Using Animations to Teach Math .....................................................................................................73
The Story of Area Formulas ..............................................................................................................75
The AIMS Model of Learning ............................................................................................................79
AREA FORMULAS
8
© 2009 AIMS Education Foundation
P
ARALLELOGRAM
UPS
UT
C
How is ﬁnding the area of a parallelogram different from ﬁnding 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 ﬁnding area of both.
ARALLELOGRAM
n
n
UT
PS
attiioo
tig
iigga
t
t
s
s
e
nvve
IIn
P
Materials
Scissors
Parallelograms
Measuring Pad
Each
group of 4 or 5
students will need
2 of each of the
parallelograms. ARALLELOGRAM
P
1. What two dimensions did you use to determine the perimeter?
U
these
are some pressing
questions.
the lengths of each pair of opposite sides
2. How did you use those dimensions to ﬁnd the perimeter?
ﬁnd the sum of all four sides
CUT UPS
3. What two dimensions do you use to determine the area of the rectangle made from the two pieces
of the parallelogram?
How is ﬁnding the area of a parallelogram different from ﬁnding the area of a
rectangle?
find the perimeter
by adding up the
four sides.
C
base and height
A
1.
Make sure
Use the Measuring Pad to ﬁnd and record the lengths of
B
the sides of each parallelogram. (Round to the nearest
students align
A
whole centimeter, if necessary.)
C
each side to the
Determine and record the perimeter of each parallelogram.
D
ruler as they
measure.
2.
Make the long side of each parallelogram the base.
D
B
C
measure the
base...
...and the
height.
4. How are the dimensions used for area different than the ones used for perimeter?
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.
5. Does it matter which side is the base?
Measure and record the length of the base and the height.
Area
is the
product
of base and
height. See
that students
recognize the
difference
between
height and
side.
No, as long as you measure the height perpendicular from the base
you chose.
Make the short side of each parallelogram the base.
Measure and record the length of the base and the height.
3.
cut and arrange
the parallelograms
into rectangles.
Cut a dotted line marking one of the heights of the parallelogram.
Cut different lines on each pair of congruent parallelograms.
Area = base • height, A = b • h
Make a rectangle using the two pieces from each parallelogram.
Determine and record the area of each parallelogram by ﬁnding
the area of its two pieces making the rectangle.
1.
Short
Long Perimeter
Side (cm) Side (cm)
(cm)
Parallelogram
A
8
12
40
Parallelogram
B
9
12
42
Parallelogram
C
12
16
56
AREA FORMULAS
Base
(cm)
Height
(cm)
Area
(cm2)
Long Base
12
8
96
Short Base
8
12
96
Long Base
12
9
6
8
72
72
16
12
9
12
144
144
2.
Short Base
Long Base
Short Base
12
6. Write a formula to describe how you use these two dimensions to get the area (A) of the
parallelogram.
AREA FORMULAS
3.
13
© 2009 AIMS Education Foundation
Confirm
that students
recognize that
the areas of the
parallelogram and
the corresponding
rectangle are
equal.
© 2009 AIMS Education Foundation
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.
ics Students reinforce their understanding of the relationship of a parallelogram to a
m
Co
rectangle when ﬁnding perimeter and area.
AREA FORMULAS
9
© 2009 AIMS Education Foundation
PARALLELOGRAM
UT U PS
C
cut out the
parallelograms
along the bold
lines.
your group
will need two
of each
parallelogram.
C
A
B
AREA FORMULAS
10
© 2009 AIMS Education Foundation
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
Measuring Pad
10 11 12 13 14 15 16 17 18 19 20 21 22
centimeters
BASE
© 2009 AIMS Education Foundation
11
AREA FORMULAS
centimeters
HEIGHT
P
ARALLELOGRAM
UT
C
UPS
How is ﬁnding the area of a parallelogram different from ﬁnding the area of a
rectangle?
1.
A
find the perimeter
by adding up the
four sides.
D
B
B
A
C
Determine and record the perimeter of each parallelogram.
C
D
Use the Measuring Pad to ﬁnd and record the lengths of
the sides of each parallelogram. (Round to the nearest
whole centimeter, if necessary.)
measure the
base...
...and the
height.
2.
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 and arrange
the parallelograms
into rectangles.
3.
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 ﬁnding
the area of its two pieces making the rectangle.
1.
2.
Short
Long Perimeter
Side (cm) Side (cm)
(cm)
Height
(cm)
Area
(cm2)
3.
Long Base
Parallelogram
A
Short Base
Long Base
Parallelogram
B
Short Base
Long Base
Parallelogram
C
AREA FORMULAS
Base
(cm)
Short Base
12
© 2009 AIMS Education Foundation
P
ARALLELOGRAM
UT
C
1. What two dimensions did you use to determine the perimeter?
UPS
these
are some pressing
questions.
2. How did you use those dimensions to ﬁnd 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.
AREA FORMULAS
13
© 2009 AIMS Education Foundation
KEEP GOING
We only needed to
measure two sides
because the sides
that are across
from each other
are the same
length.
Yeah, we measured
the sides. We used
this pad thing to
measure them.
It was
pretty far.
The perimeter
is how far it is
around the
parallelogram.
After measuring,
we added up the lengths
of the four sides to find
the perimeter.
Okay, how many
sides did you have
to measure?
After you cut out the
parallelograms at
the beginning of this
activity, you measured
some lengths. Let’s
talk about that.
4. How do you find the height
of a parallelogram?
3. What is the base of a
parallelogram?
ESSENTIAL MATH SERIES
0
1
2
3
4
5
6
7
8
12
C
12 + 16 + 12 + 16 = 56
Like for
parallelogram C,
it was 12 plus 16
plus 12 plus 16.
that’s 56.
16
9
Measuring Pad
I just
multiplied the
long side and the
short side each
by 2 and then
added them!
What else
did you do?
We call
them opposite
sides. So, we can
say that opposite
sides of a
parallelogram
have equal
length.
10 11 12 13 14 15 16 17 18 19 20 21 22
centimeters
BASE
C
That’s right,
mark. For every
parallelogram, the
sides across from each
other are not only
parallel, but they
are also equal
in length.
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
The measuring pad was like two rulers that
were perpendicular to each other. It was
easy to put one side of a parallelogram on
one of the rulers and measure it.
5. What is the formula for finding the area of a parallelogram?
2. How do you find the perimeter
of a parallelogram?
1. What do you know about the sides
of a parallelogram?
THINGS TO LOOK FOR:
Parallelogram Cut-Ups
centimeters
HEIGHT
1
12
C
Well done.
Either way
is fine for finding
the perimeter.
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22
centimeters
BASE
C
0
1
2
3
4
5
I get it, the
parallelogram is
taller when it’s
resting on the short
side, and it’s shorter
when it’s resting on
the long side.
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
6
7
8
9
Well, yeah,
red, that’s
pretty
obvious.
10 11 12 13 14 15 16 17 18 19 20 21 22
centimeters
BASE
Like if parallelogram C is resting on the
long side, then that’s the base and that’s 16. If
you measure the height from that base it’s 9.
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
using the pad, we rested one of the sides of the
parallelogram on the bottom ruler and measured
the height of the perpendicular dotted line.
16
2 • 12 + 2 • 16 = 56
The perimeter for
parallelogram C
was 2 times 12
plus 2 times 16.
centimeters
HEIGHT
AREA OF PARALLELOGRAMS, TRIANGLES, AND TRAPEZOIDS
centimeters
15
HEIGHT
AREA FORMULAS
C
© 2009 AIMS Education Foundation
So, when
parallelogram C is
resting on the short
side, the height is
12. Now we know
how tall it is!
For each
base there is
a height that
goes with it.
Hold on a
minute, vanessa,
that is a very good
observation.
Yeah, Red,
things change
when you use
a different side
for the base.
Wait, now the
height changed.
It’s not 12
anymore?
Yeah, and whatever side
the parallelogram
is resting on, we
call that the
base.
Well, we first
had to measure
the heights of the
parallelograms.
The next thing
we did was to find
the area of each of
the parallelograms,
right?
2
3
4
16
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22
centimeters
BASE
0
1
2
3
4
5
A
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22
centimeters
BASE
So, class, how
did we do it? How did
we figure out the
formula for finding
the area of any
parallelogram?
But right now,
we can use base
times height to help
us figure out a
formula for the
area of any
parallelogram.
12
8
If 12 is the base of the rectangle, then
the height is 8, right? And the area is base times
height, that’s 12 times 8 or 96.
5
0
2
0
1
1
0
2
1
3
4
5
6
7
8
9
10
11
12
13
14
15
16
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
What we found out was that length and width
on a rectangle are the same thing as base
and height. So, a rectangle has a base and
height just like all the rest of the
parallelograms.
And I wanted
to show you how
that helped us find
a formula for finding
the area of any
parallelogram.
Not really, redmond.
You’ll usually think
about the area of a
rectangle as length
times width.
And the
parallelogram
and the rectangle
have the same area
because they’re both
made out of the
same pieces.
We just cut up
the parallelogram
and put it back
together to make
a rectangle.
It was fun!
Does that
mean we have to
know two different
formulas for the area
of a rectangle?
We found
out that another
way to think about
the formula for area
of a rectangle is
that it’s base
times height.
That’s
exactly right,
Mark.
Yeah, we already knew
that the formula for
finding the area of a
rectangle is length
times width.
3
16
16
9
That’s right,
juana. You just have
to be sure that you
measure the height
from that base.
Is that the
formula for the area
of a parallelogram?
Is it base times height
for every
parallelogram?
The same thing
happens when you cut
up parallelogram B.
9
C
16
16
So, if base times
height tells us the
area of the rectangle,
then base times height
tells us the area of
the parallelogram
as well!
The area of the
rectangle is base
times height, or 9
times 16, same as the
parallelogram.
8
9
9
9
B
8
b
But for a
parallelogram the
area is just base
times height.
h
For every
parallelogram, if you
measure one of the sides, that’s
called the base. And if you then
measure the height from that
base, then the area is base
times height.
8
I think I’ve
got it! The area of
the rectangle is length
times width or base
times height! They mean
the same thing,
right?
It is,
redmond.
B
B
You can turn the parallelogram into a
rectangle and both shapes have the same base
and height. They also have the same area.
9
9
Like for parallelogram C, if we cut it up into
two pieces, we can move that triangle to the
other side and it makes a rectangle.
C
It’s because
we already knew
the formula to find
the area of a
rectangle.
C
Class, there is an
important reason
why we included the
rectangle along
with the other two
parallelograms.
centimeters
HEIGHT
centimeters
HEIGHT
C
AREA FORMULAS
A
A
© 2009 AIMS Education Foundation
That is an excellent
summary, redmond.
You’ve got it!
And you can
use the short side
for the base or you
can use the long
side, right?
So, if the area of the
rectangle is base
times height, then
the area of the
parallelogram is
base times height
as well.
4