Pythagorean Theorem and 3-D Figures

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Pythagorean Theorem and 3-D Figures
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Rectangular solids

• Every rectangular solid has
• 6 rectangular faces
• 1.
• 2.
• 3.
• 4.
• 5.
• 6.

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• 12 edges
• 1. 2.
• 3. 4.
• 5. 6.
• 7. 8.
• 9. 10.
• 11. 12.

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• 4 diagonals
• 1.
• 2.
• 3.
• 4.
• A cube is a rectangular solid in which all edges
  are congruent. All faces are squares!

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Regular Square Pyramid

• KLQP is a square and is called the base.
• J is the vertex.
• is the altitude of the pyramid and is
  perpendicular to the base at its center.
• is called the slant height and is
  perpendicular to a side of the base.

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Example 1 This box is a rectangular solid.

• If BC 3, what is AD?
• If AB 4 what is DB?
• Must use BC 3 to find
• If CG 12, what is DH?
• What is BH? AG?
• Use DB 5

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Example 2

• 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
• Use DH to find AH

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Example 2

• 2nd possibility
• Find AH
• To find AH first must find DH
• Use DH to find AH

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Example 2

• 3rd possibility
• Find AH
• To find AH first must find DH
• Use DH to find AH

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Example 3

• The square pyramid has altitude , slant
  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
• PS 12

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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

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In review,

• 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.