The Volume of Square Pyramids

1
The Volume of Square Pyramids

• By
• Monica Ayala

2
What is a square pyramid?

• A square pyramid is a pyramid whose base is
• you guessed it,
• a square.
• The height is the length from
• the apex to the base.

3
Volume of a square pyramid

• The formula for the volume of a square pyramid is
  

V 1 hb2
3
Where h is the height, and b is the length of the
base.

• But where does it come from?

4
Deriving the volume formula

• First, recall the volume of a cube is
• V b3, where b is the length of one side of the
  cube.

5
Deriving the volume formula

• Next, we figure out how many square pyramids
• (that have the same base as the cube) fit inside
  the cube.

6
Deriving the volume formula

• One fits in the bottom. (1)

7
Deriving the volume formula

• One fits in the bottom.(1)
• Another on top.(2)

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Deriving the volume formula

• One fits in the bottom.(1)
• Another on top.(2)
• One on the right side.(3)

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Deriving the volume formula

• One fits in the bottom.(1)
• Another on top.(2)
• One on the right side.(3)
• Another on the left.(4)

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Deriving the volume formula

• One fits in the bottom.(1)
• Another on top.(2)
• One on the right side.(3)
• Another on the left.(4)
• One on the far back. (5)

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Deriving the volume formula

• One fits in the bottom.(1)
• Another on top.(2)
• One on the right side.(3)
• Another on the left.(4)
• One on the far back. (5)
• Another in front. (6)

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Deriving the volume formula

• So, we can fit a total of 6 pyramids inside the
  cube.
• Thus, the volume of one pyramid is the
• volume of the cube

1
6
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Deriving the volume formula

• Now, our formula for the volume of one pyramid is

V b3
6

• that is, the volume of the cube divided by 6.

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Deriving the volume formula

• Now, this formula ? works only because we can
  fit 6 pyramids nicely in the cube, but

V b3
6
What if the height of the pyramid makes it
impossible to do this?
Maybe its taller!!
Or shorter!!
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Deriving the volume formula

• We need to find a way to integrate the variable
  for the height into our formula. ?

V b3
6
h
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Deriving the volume formula

• Observe that we can fit two pyramids across the
  height, length, or width of the cube.
• This means that the height of one pyramid is ½
  the length of b
• In other words, 2h b.

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Deriving the volume formula

• So, 2h b.
• Now, substitute this value in our formula.

This is the original formula!!!!
?
1hb2
3
V b3
6
(2h)b2
6