1. The Pythagorean Theorem in 3D
Space Diagonal
The Pythagorean Theorem in 3D

2. A rectangular prism has 4 space diagonals.
In a rectangular prism, a space diagonal is a line that goes from a vertex of the prism, through the center of the prism to the opposite vertex.
aka, ‘stick in the box’
A rectangular prism has 4 space diagonals.

3. Side c is the space diagonal.
The Pythagorean Theorem is useful when we need to find the length of a space diagonal in a rectangular prism.
Side c is the space diagonal.

4. Finding The Length Of A Space Diagonal.
Look at the rectangular prism below:
It is our task to calculate the length of the space diagonal AG which runs through the space inside the rectangular prism. F
28cm
12cm
16cm A B C D E G H
Start by considering the triangle ABC.
(2) Calculate the length of AC. A C 28 12 B
AC ≈ 30.5cm

5. Now look at the box again.
(3) Now consider triangle ACG.
(4) Calculate the length of AG. F
28cm
12cm
16cm A B C D E G H
30.5cm A G
30.5 16 C
AG =
AG = 34.4cm

6. OR OR OR OR OR OR OR OR OR

7. By the Pythagorean Theorem, l2 + w2 = x2 and x2 + h2 = d2.
A diagonal of a three-dimensional figure connects
two vertices of two different faces.
Diagonal d of a rectangular prism is shown in the diagram.
By the Pythagorean Theorem,
l2 + w2 = x2 and x2 + h2 = d2.
Using substitution, l2 + w2 + h2 = d2.

8. Pythagorean Theorem 2D vs 3Dc b d

9. The Space Diagonal
l2 + w2 + h2 = d2.

10. Space Diagonal Pythagoras In 3D Shapes.F
28cm
12cm
16cm A B C D E G H

11. What Goes In The Box ? Calculate the lengths of the space diagonals in
the rectangular prisms below:
(1) 4m 3m
12m
(2)
26cm
17cm
23cm

12. Using the Pythagorean Theorem in Three Dimensions
Find the length of the space diagonal of a
6 cm by 8 cm by 10 cm rectangular prism
Substitute
Simplify

13. Using the Pythagorean Theorem in Three Dimensions
Find the height of a rectangular prism with a 12 in. by 7 in. base and a space diagonal of 15 in.
Substitute
Square both sides of the equation.
225 = h2
Simplify.
h2 = 32
Solve for h2.
Take the square root of both sides.