Volume of a tapering rectangular solid

Zachary Hoyt · Mar 4, 2014

I have been researching pier type foundations and am thinking of building forms to cast piers that would be 5 feet high with one foot above ground and would taper from 22x22 at the bottom to 8x8 at the top. I looked at the sonotubes with feet and various other methods, but the tapered column could be formed with willow lumber which I can produce very cheaply and which is not good for much else. Our Amish neighbors poured a tapered top like this over a round 24" footer and I talked with another neighbor who had put 55 gallon drums in the ground and filled them with concrete. I am wondering if there is a formula for finding the volume of this shape, sort of like the Washington Monument but without the pointy bit on top. Any advice will be much appreciated. I am not even sure if there is a name for this shape or not, if there is I either never learned it or have forgotten.
Zach

Billy NY · Mar 4, 2014

Formula for the volume of a trapezoid ?
Volume of Trapezoid

TAS · Mar 4, 2014

The technical term for this would be Frustum of Pyramid. I got this formula from the Machinery's Handbook, if you have one of those lying around.
Formula:
V= h/3 * (A1 + A2 + (square root of A1 + A2))

A1 being the area of the top (8x8), and A2 being the area of the bottom (22x22)

Adirondack case guy · Mar 4, 2014

Redneck Calculation.
The bottom, 22"square = 3.36 sq.ft.
The top. 8" square =.444 sq.ft.
Add them together and divide by 2,= 1.90
Multiply that by 5' height of pier, = 9.51 cu.ft.
Divide 9,51 x 27, = .35 yrds of concrete per pier.
Loren, the Acg.

TAS · Mar 4, 2014

Correction the A1 + A2 under the square root should be A1 * A2.

So the correct formula is:
V = h/3 * (A1 + A2 + (sq root of (A1 * A2)))

= 60"/3 * (64 sq in + 484 sq in + (sq rt (64 sq in * 484 sq in))

= 14480 cu in. = 8.4 cu ft.

definitely check my math on that

Zachary Hoyt · Mar 4, 2014

Thanks, I can figure a trapezoid but where this is a trapezoid both ways it had me wondering.
Zach

Zachary Hoyt · Mar 4, 2014

Thank you, that is very helpful and now I have learned a new word too.
Zach

Zachary Hoyt · Mar 4, 2014

Thanks, that is kind of how I had made my estimate. I knew that method would work for a trapezoidal solid but I didn't know if it tapering both ways would throw it off somehow. It looks like it was pretty close to the scientific method, and of course with concrete there is always some error factor depending on the ground underneath and just how straight the forms are and such.
Zach

HillsofTn · Mar 4, 2014

I think what you are looking for is the "volume of a truncated square pyramid". If so, google that expression and it will give you a nice simple formula based on the three known measurements. 1/3(22squared + 22 times 8 +8squared)times 60. That would give you the volume in cubic inches, which divided by 1,728 would convert the answer to cubic yards.

(But then, it is has been a long time since I stayed at a Holiday Inn - the magic may have worn off!)

HillsofTn · Mar 4, 2014

Oops, not cubic yards - cubic feet! (I type like I have lived, fast with a lot of mistakes!)

sms · Mar 4, 2014

Tapered battered pier as you described is best as frost can't push on anything as lifts. That's the way I would do it.

Zachary Hoyt · Mar 4, 2014

Thanks, that calculator was very handy. It said 8.38 CF per pier when I put the numbers in. I should have learned by now not to underestimate Google.
Zach

Zachary Hoyt · Mar 4, 2014

Thanks, that's good to hear. It's been rolling around in my mind the last few weeks as I try to figure out a plan and that was the best idea I could come up with that I could form with lumber instead of having to buy something.
Zach

Volume of a tapering rectangular solid

Zachary Hoyt

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Billy NY

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TAS

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Adirondack case guy

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TAS

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Zachary Hoyt

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