Using the Pythagorean Theorem in
3-Dimensional Shapes
Focus 5 - Learning Goal #1: Students will understand
and apply the Pythagorean Theorem.
4
3
2
1
0
In addition to level
3.0 and beyond
what was taught in
class, the student
may:
 Make
connection
with other
concepts in
math.
 Make
connection
with other
content areas.
 Explain the
relationship
between the
Pythagorean
Theorem and
the distance
formula.
The student will understand
and apply the Pythagorean
Theorem.
 Prove the Pythagorean
Theorem and its
converse.
 Apply the Pythagorean
Theorem to real world
and mathematical
situations.
 Find the distance
between 2 points on a
coordinate plane using
the Pythagorean
Theorem.
The student will
understand the
relationship between the
areas of the squares of the
legs and area of the
square of the hypotenuse
of a right triangle.
 Explain the
Pythagorean
Theorem and its
converse.
 Create a right
triangle on a
coordinate plane,
given 2 points.
With help from the teacher,
the student has partial
success with level 2 and
level 3 elements.
 Plot 3 ordered pairs to
make a right triangle
 Identify the legs and
the hypotenuse of a
right triangle
Find the distance between 2
points on the coordinate
grid (horizontal and vertical
axis).
Even with
help,
students
have no
success with
the unit
content.
Pythagorean Theorem Review
 The Pythagorean Theorem states: in any right triangle,
the square of the hypotenuse is equal to the sum of the
squares of the other two sides. a2 + b2 = c2
c
b
 Find the length of AC in the
diagram below:
a2 + 52 = 132
a2 + 25 = 169
a2 = 144
a = 12
a
3-Dimensional Figures
12 m
5m
 How would you find the length of segment AV?
 Do you see a right triangle inside the shape?
 How would you find the length of segment AF?
 Do you see the right triangle inside the shape?
Use the Pythagorean Theorem to find
the length of diagonal AV.
 To find AV we need the lengths of
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sides AM and MV.
What is the length of MV? 12 m
12 m
What is the length of AM? 5 m
5m
We have enough information to
solve the problem.
a2 + b2 = c2
52 + 122 = c2
25 + 144 = c2
169 = c2
13 = c
The length of diagonal AV is 13 m.
Use the Pythagorean Theorem to find
the length of diagonal AF.
G
 AF is the diagonal going through a prism. To
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find AF, we use the Pythagorean Theorem
6cm
differently.
To find AF we need to know the length, width and height
of the prism.
What is the length = AB? 6 cm
What is the width = FG? 2 cm
What is the height = GB? 3 cm
(length)2 + (width)2 + (height)2 = (diagonal of prism)2
6 2 + 22 + 3 2 = d 2
36 + 4 + 9 = d2
49 = d2
7=d
The length of the diagonal AF is 7 cm.
Use the Pythagorean Theorem to find
the length of diagonal TX.
 TX is the diagonal going through a prism.
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To find TX, we use the Pythagorean
Theorem differently.
What is the length = TU? 12 in
What is the width = XW? 6 in
What is the height = UW? 9 in
(length)2 + (width)2 + (height)2 = (diagonal of prism)2
122 + 62 + 92 = d2
144 + 36 + 81 = d2
261 = d2
The length of the diagonal TX is 𝟐𝟔𝟏
261 = d
which is between 16 and 17 inches.