## Jhannybean one year ago How do I prove that the volume of a rectangular pyramid is divided by 3? @Zarkon

1. jim_thompson5910

2. Jhannybean

What I've got so far is that.. |dw:1444608594950:dw|

3. Jhannybean

So when I was thinking about it... the sides of a rectangular prism make half of a pyramid, therefore if we combined them together that would give us 1 full pyramid, and then the sides of the rectangular prism make 2 more triangular prisms altogether equalling 3. Does my method make any sense to anyone? And thanks Jim! I was trying to figure it out on my own,this is how far I've gotten :) lol.

4. amistre64

|dw:1444609095726:dw|

5. amistre64

$\int_{0}^{h}xy~dz$ $\int_{0}^{h}kn~z^2~dz$ $\frac13 kn~h^3~$ pfft, somethings aloof

6. Jhannybean

Ahh... I was approaching this without calculus haha

7. amistre64

i spose h=zmax in this case ... $\frac{1}{3}(kz)(nz)z=\frac{1}{3}xyz$ might have a bad change of variable is all

8. Jhannybean

How did you find $$x=kz$$ and $$y=nz$$? are $$k$$ and $$n$$ just scaling factors?

9. amistre64

to make life simpler i turned the pyramid upside down so that the lines defined would go thru the origin is all

10. amistre64

|dw:1444609883979:dw| allowing k and n to account for any width/depth and z for the height

11. amistre64

the volume of any slice being xy dz, or dh for a dummy variable

12. Jhannybean

I'll bbs. afking for a moment.

13. mathstudent55