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 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!!!! 