The volume of a pyramid with height h and square base of side length a is V = 1/3ha^2.

Use this fact to prove that the volume of a truncated pyramid is V=1/3h(a^2 + ab + b^2), where h is the height and b and a are the lengths of the sides of the square top and bottom, respectively.

Question

The volume of a pyramid with height h and square base of side length a is V = 1/3ha^2.

Use this fact to prove that the volume of a truncated  pyramid is V=1/3h(a^2 + ab + b^2), where h is the height and b and a are the lengths of the sides of the square top and bottom, respectively.

anonymous · Answer

I found an explanation online but part of it confuses me. It's the part with the explanation of the formula of the whole pyramid. Any assistance would be appreciated. http://mathforum.org/kb/message.jspa?messageID=1085689&tstart=0

anonymous · Answer

They are setting up the problem such tat thhat the heights of the parts (the truncated part and the removed part are H and M, respectively..

That being the case, if you had a whole and complete pyramid (botht he truncated and the removed parts fused back together), the entire height would be H + M.

anonymous · Answer

So, given your volume formula, the volume of the whoel and complete pyramid would be$$V=(1/3)(H+M)a^{2}$$

anonymous · Answer

Earlier, they worked out that M=bH/(a - b)...I assume you understood that part since it came before where you said you got lost.  If you don't understand, let me know and I can explain it.

Plugging this into the above formula in place of M gives us$$V=(1/3)(H+\frac{bH}{a-b})a^{2}$$

anonymous · Answer

I'm going to break for a moment here and say that their explaination has errors in it...so don't go too much with them.  They missed in that last step the a^2...so I'd just continue from here if I were you.

anonymous · Answer

ok I get that part. so how would I get from that last equation you posted to the next (H/3)*(B*H/(A-B) + A^2)? or did they make a mistake there too?

anonymous · Answer

I just worked it long had...and it's a bit of arithmetic.  I probabily took the long way around, but I eventually got there.

It would probably be easier if I type it into a word doc and posted if that's alright by you.  The equation editor here is a little tempermental.

anonymous · Answer

That would be great! thanks!

anonymous · Answer

Give me a minute to type it in then :)

anonymous · Answer

Ok...here it is.

anonymous · Answer

Thank you so much! that really cleared everything up :)