1. Using the Pythagorean Theorem in 3-Dimensional Shapes

2. 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
Find the length of AC in the diagram below: c a b
a = 132
a = 169
a2 = 144
a = 12

3. 3-Dimensional Figures 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?

4. To Find the Diagonal of a 3D Prism
3D Figures
To Find the slant height of a cone or pyramid
It doesn’t matter that it is 3D, there is a right triangle hidden in the problem
The slant height is just the hypotenuse of a right triangle.
Use a2 + b2 = c2
To Find the Diagonal of a 3D Prism
Diagonals of prisms have a different formula
(length)2 + (width)2 + (height)2 = (diagonal of prism)2

5. Use the Pythagorean Theorem to find the length of diagonal AF.
AF is the diagonal going through a prism. To find AF, we use the Pythagorean Theorem differently.
To find AF we need to know the length, width and height of the prism (because it is a 3D shape, we need 3 measurements)
What is the length = AB?
What is the width = FG?
What is the height = GB?
(length)2 + (width)2 + (height)2 = (diagonal of prism)2
= d2
= d2
49 = d2
7 = d
6cm
6 cm
2 cm
3 cm
The length of the diagonal AF is 7 cm.

6. Use the Pythagorean Theorem to find the length of diagonal TX.
TX is the diagonal going through a prism. To find TX, we use the Pythagorean Theorem differently.
What is the length = TU?
What is the width = XW?
What is the height = UW?
(length)2 + (width)2 + (height)2 = (diagonal of prism)2
= d2
= d2
261 = d2
261 = d
12 in
6 in
9 in
The length of the diagonal TX is 𝟐𝟔𝟏 which is between 16 and 17 inches.