Pythagorean Theorem and 3D Figures
Rectangular solids
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Every rectangular solid has:
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6 rectangular faces:
1. re c ta n g le A B H G
2. re cta n g le B D F H
3. re c ta n g le D C E F
4. re c ta n g le C A G E
5. re c ta n g le C D B A
6. re cta n g le E F H G
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12 edges:
1. A C
3. D B
5. A G
7. H F
9. E G
11. D F
2.
4.
6.
8.
10.
12.
CD
BA
GH
FE
BH
CE
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4 diagonals:
1. A F
2. C H
3. D G
4. BE
A cube is a rectangular solid in which all
edges are congruent. All faces are squares!
Regular Square Pyramid
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KLQP
is a square and is called the base.
J
is the vertex.
J M is the altitude
of the pyramid and is
perpendicular
to the base at its center.
J N is called the slant height and is perpendicular to
a side of the base.
Example 1: This box is a rectangular solid.
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If BC = 3, what is AD? A D  3
If AB = 4 what is DB?
Must use BC = 3 to find
2
2
3  4  DB
2
DB  5
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If CG =12, what is DH?
D H  12
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What is BH? AG?
Use DB = 5
2
2
2
5  12  BH
B H  13
A G  13
Example 2:
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The dimensions of a rectangular solid are 3, 5, and 7.
Find the diagonal.
Draw a rectangular solid.
Does it matter where the numbers go?
1st possibility
Find AH
To find AH first must find DH
2
3 7
2
DH 
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 DH
2
58
Use DH to find AH

58

2
AH 
2
 5  AH
83
2
Example 2:
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2nd possibility
Find AH
To find AH first must find DH
2
2
3  5  DH
DH 
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2
34
Use DH to find AH

34

AH 
2
7
2
83
 AH
2
Example 2:
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3rd possibility
Find AH
To find AH first must find DH
2
2
7  5  DH
DH 
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2
74
Use DH to find AH

74

AH 
2
2
 3  AH
83
2
Example 3:
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PR
The square pyramid has altitude
, slant PS
height
,
perimeter of JKMO = 40, and PK = 13.
Find
a.) JK
Since it is a square pyramid,
the base is a square
A square has 4 congruent sides
If the perimeter is 40 then each
side is 10 so JK = 10
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b.) PS
In a square pyramid, the slant
height bisects the base edge
So KS = 5
Use PK = 13 to find PS
2
2
13  5  PS
PS = 12
2
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C.) PR
In a square pyramid, the altitude
height bisects the base edge
So RS = 5
Use PS = 12 to find PR
2
2
12  5  PR
PR 
119
2
In review,
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With any type of solid, you need to create
right triangles that you can use the
Pythagorean to find the missing length.
You may need to use the Pythagorean
theorem more than once to find the missing
length.