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not much of a brain teaser...but



This has given me a headache!

Anyone can post some kind of ratio -formula- for dividing the VOLUME of an isosceles pyramid into 3 EQUAL sections (cut horizontally)???

I reeeeeally need this.


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Let the height - apex to base - of the original pyramid be c.

Cut the pyramid with a plane parallel to the original base a distance b from the apex.

This defines a second pyramid with height b.

Cut again, a distance a from the apex. This defines a third pyramid, of height a.

Volume of any pyramid is proportional to the product of base area and height [1/3 i think - doesn't matter]

Area of pyramid bases is proportional to the square of their heights

So, volume of pyramids is proportional to cube of their heights.

You want the pyramid volumes to be in the ratio of 1[a] : 2(B) : 3[c]

so that the slices will have equal volumes.

a[cubed] = 1/2 b[cubed] = 1/3 c[cubed].

That should give you the ratios you need.

Caveat, I did this on the fly, and I'm only half awake,

so maybe some other genius will come along and correct this.

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