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 hb 2 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 = b 3, where b is the length of one side of the cube. b b b

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 pyramids inside the cube.  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

13. Deriving the volume formula  Now, our formula for the volume of one pyramid is: V= b3b3b3b3 6  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… V= b3b3b3b3 6 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.  V= b3b3b3b3 6 h

16. Deriving the volume formula OOOObserve that we can fit two pyramids across the height, length, or width of the cube. TTTThis means that the height of one pyramid is ½ the length of b IIIIn other words, 2h = b.

17.  So, 2h = b.  Now, substitute this value in our formula. = 1hb 2 3 V= b3b3b3b36 = (2h)b 26 This is the original formula!!!! 