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Triangles
Surrounded by Squares
If you know the lengths of the sides of a triangle, how can you determine if the
triangle is acute, right, or obtuse?
This activity introduces students to the Pythagorean relationship. Students build triangles using the sides
of squares as the sides of the triangle. Since there is a square on each side of the triangle they just built,
they can compare the sum of the areas of the two smallest squares to the area of the largest. By looking at
a variety of triangles and comparing the sum of the areas of the squares on the two smallest sides to the
area of the square on the biggest side, they come to see that if the sum of the areas of the two smallest
squares is equal to the area of the largest, then the triangle is a right triangle. If the sum of the areas of
the two smallest is not equal to the largest, then the triangle is either acute or obtuse. Discovering these
equalities and inequalities helps students understand that the Pythagorean relationship is true only for
right triangles.
n
n
o
o
i
i
t
t
a
a
ttiigg
s
s
e
e
nvv
IIn
tig
Triangles
make sure
the square
corners touch
to make the
triangle
corners
• For each set of three numbers, find squares that have sides
equal to these numbers.
• Build a triangle so that one side of each square is a side of
the triangle.
• On a blank sheet of paper, glue or tape the squares to that
you have record of each triangle.
• Fill in the chart for each triangle.
Each group
of Students
needs the two
pages of squares
to make all seven
triangles. They can
work together
and share
the finished
triangles.
Length of Sides
a
3
b
6
c
8
Area of Squares
a2
9
Comparison of Areas
<
=
c2
>
b2
36
Kind of Triangle
obtuse
right
acute
a2+b2
If
45
<
64
then the
triangle is
obtuse
right
3
6
4
8
5
9 16
If
25 = 25
then the
triangle is
3
4
6
5
8
6
16 25
If
41 > 36
then the
triangle is
acute
3
4
6
8
10
8
1664
If
80 < 100
then the
triangle is
obtuse
right
12
6 13
8
25144
If
169 = 169
then the
triangle is
10
3 12
6 13
8
100 144
If
244 > 169
then the
triangle is
acute
3
6
36 64
If
100 = 100
then the
triangle is
right
3
5
6
8
10
8
1. How does a2 + b2 compare to c2 for those triangles that are obtuse?
Encourage
the students
to answer these
both in words and
with an inequality
or equation.
When
students
make the
triangles, Be
sure the corners
of the squares
form the
corners of the
triangle.
Surrounded by Squares
Materials
Scissors
Tape or glue
Blank paper
Student pages
a2 + b2 is less than c2
2. How does a2 + b2 compare to c2 for those triangles that are right?
They are equal.
3. How does a2 + b2 compare to c2 for those triangles that are acute?
a2 + b2 is greater than c2
THE PYTHAGOREAN RELATIONSHIP
10
1
If
students
use different
colorED
highlighters or
colored pencils
and shade in all
the same type of
triangle and the
matching inequalities
with one color,
they will notice
the relationship
between the
areas more
EASILY.
There is an openended extension included
that you can use to help
students transfer their
learning or to assess their
understanding.
© 2009 AIMS Education Foundation
© 2011 AIMS Education Foundation
Triangles
Surrounded by Squares
make sure
the square
corners touch
to make the
triangle
corners
• For each set of three numbers, find squares that have sides
equal to these numbers.
• Build a triangle so that one side of each square is a side of
the triangle.
• On a blank sheet of paper, glue or tape the squares to that
you have record of each triangle.
• Fill in the chart for each triangle.
Length of Sides
Area of Squares
Comparison of Areas
<
a2+b2 =
c2
>
Kind of Triangle
obtuse
right
acute
a
b
c
a2
b2
3
6
8
9
36
3
6
4
8
5
If
then the
triangle is
3
4
6
5
8
6
If
then the
triangle is
3
4
6
8
10
8
If
then the
triangle is
3
5
12
6 13
8
If
then the
triangle is
10
3 12
6 13
8
If
then the
triangle is
3
6
If
then the
triangle is
6
8
10
8
If
45
<
64
then the
triangle is
obtuse
1. How does a2 + b2 compare to c2 for those triangles that are obtuse?
2. How does a2 + b2 compare to c2 for those triangles that are right?
3. How does a2 + b2 compare to c2 for those triangles that are acute?
2
© 2011 AIMS Education Foundation
Triangles
Surrounded by Squares
13
5
2
5
2
5
2
12
2
8
2
8
2
10
8
3
3
2
2
2
3
2
Cut along
the dotted lines
to get
the squares
2
© 2011 AIMS Education Foundation
Triangles
Surrounded by Squares
Cut out the
squares by
cutting along
the dotted
lines
6
4
2
10
13
2
4
2
6
2
6
2
4
2
10
2
2
12
4
2
2
© 2011 AIMS Education Foundation
Triangles
Surrounded by Squares
Extension
How can you decide what size squares you need to make an obtuse, right, or acute triangle?
This extension of the investigation provides an opportunity to assess how well students understand and
can apply what they are learning. Students are given five different-sized squares and are asked to determine which ones to use to make each type of triangle. Students should work backwards by calculating the
area of each square and then finding which combinations create the needed equalities and inequalities.
n
n
o
o
i
i
t
t
iiggaa
t
t
s
s
e
nvve
IIn
tig
Triangles
HELP!
I’m surrounded
by squares!
Surrounded by Squares
Extension
Materials
Scissors
Student page
Use the chart to determine which three squares you could use to make an obtuse, right, and acute
triangle. Then cut out the squares and confirm your choices.
Length of Sides
Area of Squares
a2
Comparison of Areas
<
=
c2
>
b2
Kind of Triangle
obtuse
right
acute
a2+b2
a
b
c
3
6
8
If
<
then the
triangle is obtuse
3
6
8
If
then the
triangle is
3
6
8
If
=
>
To solve
this, students
will need to
decide if they are
looking for a sum
of areas that is
less than, equal
to, or greater
than a single
square.
right
then the
triangle is acute
If the lengths of the sides of a triangle are represented by a, b, c, and if a2 + b2 = c2, then what kind of
triangle must it be?
a right triangle
The area of
the squares
will need to
be calculated
before trying
out different
sums.
9
2
15
8
12
2
2
Students
need to make
their predictions
with calculations
before cutting
out the squares
to confirm their
choices.
2
6
THE PYTHAGOREAN RELATIONSHIP
14
2
© 2009 AIMS Education Foundation
ics The comic shows the class discussing what they did in the investigation. It can be a
m
Co
way for your class to rehearse what they did, and perhaps clarify some things that they
have not understood.
5
© 2011 AIMS Education Foundation
Triangles
HELP!
I’m surrounded
by squares!
Surrounded by Squares
Extension
Use the chart to determine which three squares you could use to make an obtuse, right, and acute
triangle. Then cut out the triangles and confirm your choices.
Length of Sides
Area of Squares
a2
Comparison of Areas
<
a2+b2 =
c2
>
b2
Kind of Triangle
obtuse
right
acute
a
b
c
3
6
8
If
then the
triangle is obtuse
3
6
8
If
then the
triangle is
3
6
8
If
then the
triangle is acute
right
If the lengths of the sides of a triangle are represented by a, b, c, and if a2 + b2 = c2, then what kind of
triangle must it be?
9
2
15
8
12
2
2
2
6
6
2
© 2011 AIMS Education Foundation
ESSENTIAL MATH SERIES
Well,
mostly what
we did was use
squares to build
triangles.
Okay, and what happens
when the sum of the
areas of the two
smallest squares is
less than the
largest one?
Then the
triangle
is obtuse.
there were two
like that.
And when
those two areas
are equal, It makes
a right triangle.
We found two
others like
that.
Yeah, for that
triangle the areas
of the two smallest
squares add up to be
the same as the area
of the biggest
one.
Now that
you’ve finished
the activity, let’s
talk about what
you learned.
1. If you know the lengths of
the sides of a triangle, how
can you decide whether it
is acute, right, or obtuse?
THINGS TO LOOK FOR:
8
2
And when the two
smallest ones add up
to be more than the
biggest one, the
triangle is acute.
6
2
That’s the triangle
with sides that are
6, 8, and 10.
This is
one of the
triangles you
made, right?
6
2
2
Good job.
That’s exactly
what I wanted you
to discover in the
first part of
this activity.
And the
other two had
sides of 3, 4, 5
and 5, 12, 13.
8
2. What kind of triangle has squares
on two of its sides that have a
total area equal to the area of
the square on the third side?
Triangles Surrounded by Squares
2
10
2
10
THE PYTHAGOREAN RELATIONSHIP
1
I used two
of the 8 squares.
Is that okay? That
triangle looks
like it’s acute
for sure.
To make the
acute triangle, I
picked the 6 square,
the 8 square, and the
9 square.
9
2
81
8
2
And when you add the
areas of the two 8
squares you get 128,
and that’s a lot
bigger than 81.
64 + 64
24 = 128
I picked the same ones,
I cut out the squares
and put them together.
I know it works.
It’s obtuse.
2
2
8
9
It is okay
to use two
squares that
are the same,
mark. In fact,
what would
happen if all
three were
the same?
So, doesn’t
it have to be acute?
that’s the way it is
in the chart with the
other triangles
we built.
15
2
15
2
2
225
8
81 + 64 = 145
2
2
2
81
Three!?
9
Is that even
possible?
8
2
64 + 36 = 100
That’s right, vanessa,
and if you had built
that triangle, it would
look something
like this.
2
9
6
12
How could you find three of these squares
that would make an obtuse triangle? How
did you figure that out?
But I didn’t cut
out the squares to check
it. I just know that 6
squared plus 8 squared is
100 and that’s bigger
than 81.
So, I used
the 8 square,
the 9 square,
and the 15
square.
What we learned
from the first part is that
if the areas of the two
smallest squares together
is less than the biggest one,
then the triangle will
be obtuse.
There was
a second part
to the activity.
You were
given a new
set of five
squares to look
at, and you were
asked to pick three
of them that you
could use to make
a certain kind of
a triangle.
2
2
8
7
6
© 2011 AIMS Education Foundation
2
8
© 2011 AIMS Education Foundation
Well, you
could, but
you don’t
need to.
Couldn’t
we just pretend
that there are
squares on each
of the sides? then
it would be just
like if we had built
it with those
squares.
So far we have been
looking at triangles
that were built using
the sides of squares.
I did, Mr. David. I used
the 15 square, a 12
square, and a 9 square.
and it works.
Sure it is. The triangle
would have to be acute,
because all the sides
and all the angles
would be the same:
60 degrees.
225
2
CLASS, TAKE A LOOK AT THIS
TRiangle. it has sides that
are 5, 8, and 9. is it a right
triangle?
15
2
12
81 + 144
64 =2145
= 145
9 squared plus 12
squared is equal to
15 squared. So it’s
a right triangle.
60 Degrees?
that’s definitely
an acute triangle.
2
9
9
5
2
64
Okay, and how about
right triangles? Did
you find three squares
to use to build a
right triangle?
64 64
8
If
If
If
<
>
=
81
225
225
obtuse
right
acute
right
then the
triangle is acute
then the
triangle is
then the
triangle is obtuse
Kind of Triangle
Interesting.
Can you explain
what you mean by
that, shuttle?
Don’t be so sure.
Use what you’ve just
learned. Don’t depend
on how the drawing
looks.
128
225
145
Comparison of Areas
<
a2+b2 =
c2
>
Yeah, duh.
Look at it. You
can just see
that it’s a right
triangle.
8
9
3
8
6
8
81 144
b2
64 81
a2
3 12
6 15
8
9
c
Area of Squares
8
15
b
6
9
a
3
8
Length of Sides
Okay, class, this table shows us one way
that you could have picked the squares to make
the triangles. there were actually several ways
you could make the acute and the obtuse triangles,
but only one way to make a right triangle.
8
2
8
64 + 64
24 = 128
Look, if you add up
the areas of two of
the squares, it’s
twice as much as
the third one.
2
8
That’s a very
good summary of
what we’ve done in
this activity,
shuttle.
c
a
c
b
If a2 plus b2
is equal to c2,
then it’s a right
triangle.
a
a 2 + b2 = c2
Let me see
if I have this
right.
5
8
b
What if a2 + b2
is equal to c2?
What do you know
about the triangle?
what kind is it?
9
9 = 81
2
Look at this
triangle. the
lengths of the
shortest sides
are a and b.
5 2 + 82 = 89
Okay, 5 and 8
are the shortest
two sides, right?
and 5 squared
plus 8 squared is
89. That’s bigger
than 9 squared.
that means the
triangle is acute.
It’s not a
right triangle.
class,
as we go
forward…
a
c
b
a 2 + b2 > c2
If a2 and b2
is greater than
c2, then it’s an
acute triangle.
Well, I guess it’s got
to be a right triangle,
because they’re equal.
I see letters? Is this
like a forumla?
c
b
a 2 + b2 = c2
4
What’s going
to be most
important is the
connection
between right
triangles and
a
a 2 + b2 < c2
obtuse triangle.
If a2 and b2
is less than
c2, then it’s an
right triangle
a 2 + b2 = c2
Yes, vanessa, it is like
a formula. we call
this an equation.
Okay, that’s
good, shuttle.
Now let me ask
another
question.
THE PYTHAGOREAN RELATIONSHIP
ESSENTIAL MATH SERIES
Triangles Surrounded by Squares
THINGS TO LOOK FOR:
1. If you know the lengths of
the sides of a triangle, how
can you decide whether it
is acute, right, or obtuse?
Well,
mostly what
we did was use
squares to build
triangles.
2
This is
one of the
triangles you
made, right?
10
Now that
you’ve finished
the activity, let’s
talk about what
you learned.
2. What kind of triangle has squares
on two of its sides that have a
total area equal to the area of
the square on the third side?
6
2
8
Yeah, for that
triangle the areas
of the two smallest
squares add up to be
the same as the area
of the biggest
one.
And the
other two had
sides of 3, 4, 5
and 5, 12, 13.
10
2
That’s the triangle
with sides that are
6, 8, and 10.
2
And when
those two areas
are equal, It makes
a right triangle.
We found two
others like
that.
Okay, and what happens
when the sum of the
areas of the two
smallest squares is
less than the
largest one?
6
2
8
2
And when the two
smallest ones add up
to be more than the
biggest one, the
triangle is acute.
Then the
triangle
is obtuse.
there were two
like that.
Good job.
That’s exactly
what I wanted you
to discover in the
first part of
this activity.
1
9
© 2011 AIMS Education Foundation
How could you find three of these squares
that would make an obtuse triangle? How
did you figure that out?
There was
a second part
to the activity.
You were
given a new
set of five
squares to look
at, and you were
asked to pick three
of them that you
could use to make
a certain kind of
a triangle.
9
2
15
8
12
2
2
6
I picked the same ones,
I cut out the squares
and put them together.
I know it works.
It’s obtuse.
What we learned
from the first part is that
if the areas of the two
smallest squares together
is less than the biggest one,
then the triangle will
be obtuse.
2
2
81 + 64 = 145
2
8
9
So, I used
the 8 square,
the 9 square,
and the 15
square.
15
2
2
225
To make the
acute triangle, I
picked the 6 square,
the 8 square, and the
9 square.
That’s right, vanessa,
and if you had built
that triangle, it would
look something
like this.
But I didn’t cut
out the squares to check
it. I just know that 6
squared plus 8 squared is
100 and that’s bigger
than 81.
64 + 36 = 100
2
8
2
6
So, doesn’t
it have to be acute?
that’s the way it is
in the chart with the
other triangles
we built.
9
2
81
And when you add the
areas of the two 8
squares you get 128,
and that’s a lot
bigger than 81.
It is okay
to use two
squares that
are the same,
mark. In fact,
what would
happen if all
three were
the same?
64 + 64
24 = 128
2
2
8
8
I used two
of the 8 squares.
Is that okay? That
triangle looks
like it’s acute
for sure.
9
Three!?
Is that even
possible?
2
81
2
10
© 2011 AIMS Education Foundation
Sure it is. The triangle
would have to be acute,
because all the sides
and all the angles
would be the same:
60 degrees.
Okay, and how about
right triangles? Did
you find three squares
to use to build a
right triangle?
Look, if you add up
the areas of two of
the squares, it’s
twice as much as
the third one.
60 Degrees?
that’s definitely
an acute triangle.
64 + 64
24 = 128
2
2
8
8
8
2
64
I did, Mr. David. I used
the 15 square, a 12
square, and a 9 square.
and it works.
9 squared plus 12
squared is equal to
15 squared. So it’s
a right triangle.
Okay, class, this table shows us one way
that you could have picked the squares to make
the triangles. there were actually several ways
you could make the acute and the obtuse triangles,
but only one way to make a right triangle.
81 + 144
64 =2145
= 145
2
Length of Sides
9
2
12
15
2
a2
Comparison of Areas
<
=
c2
>
b2
Kind of Triangle
obtuse
right
acute
a2+b2
b
3
8
6 15
8
9
64 81
If
145
<
225
then the
triangle is obtuse
3 12
6 15
8
9
81 144
If
225
=
225
then the
triangle is
3
8
225
6
8
c
Area of Squares
a
8
9
64 64
If
128
>
81
right
then the
triangle is acute
Don’t be so sure.
Use what you’ve just
learned. Don’t depend
on how the drawing
looks.
So far we have been
looking at triangles
that were built using
the sides of squares.
Yeah, duh.
Look at it. You
can just see
that it’s a right
triangle.
9
8
5
Couldn’t
we just pretend
that there are
squares on each
of the sides? then
it would be just
like if we had built
it with those
squares.
Interesting.
Can you explain
what you mean by
that, shuttle?
Well, you
could, but
you don’t
need to.
3
11
© 2011 AIMS Education Foundation
Okay, 5 and 8
are the shortest
two sides, right?
and 5 squared
plus 8 squared is
89. That’s bigger
than 9 squared.
that means the
triangle is acute.
It’s not a
right triangle.
Okay, that’s
good, shuttle.
Now let me ask
another
question.
92 = 81
9
5 2 + 82 = 89
8
5
Look at this
triangle. the
lengths of the
shortest sides
are a and b.
What if a2 + b2
is equal to c2?
What do you know
about the triangle?
what kind is it?
Well, I guess it’s got
to be a right triangle,
because they’re equal.
I see letters? Is this
like a forumla?
Yes, vanessa, it is like
a formula. we call
this an equation.
a 2 + b2 = c2
right triangle
b
a
c
Let me see
if I have this
right.
If a2 and b2
is greater than
c2, then it’s an
acute triangle.
If a2 plus b2
is equal to c2,
then it’s a right
triangle.
a 2 + b2 = c2
a 2 + b2 > c2
b
a
b
a
c
c
That’s a very
good summary of
what we’ve done in
this activity,
shuttle.
class,
as we go
forward…
If a2 and b2
is less than
c2, then it’s an
obtuse triangle.
a 2 + b2 < c2
b
a
c
What’s going
to be most
important is the
connection
between right
triangles and
a 2 + b2 = c2
4
12
© 2011 AIMS Education Foundation