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The Volume of Square Pyramids

The Volume of Square Pyramids. By Monica Ayala. 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. Volume of a square pyramid. The formula for the volume of a square pyramid is .

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The Volume of Square Pyramids

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  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 Where h is the height, and b is the length of the base. • But where does it come from?

  4. b b b 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)

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

  9. 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)

  10. 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)

  11. 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)

  12. 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

  13. Deriving the volume formula • Now, our formula for the volume of one pyramid is: • that is, the volume of the cube divided by 6.

  14. Deriving the volume formula • Now, this formula  works only because we can fit 6 pyramids nicely in the cube, but… What if the height of the pyramid makes it impossible to do this? Maybe it’s taller!! Or shorter!!

  15. Deriving the volume formula • We need to find a way to integrate the variable for the height into our formula.  h

  16. 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.

  17. Deriving the volume formula • So, 2h = b. • Now, substitute this value in our formula. This is the original formula!!!! 

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