Frustum Pyramid

Scrutch

New member
Joined
Jun 30, 2005
Messages
14
Hi all,

Well, as you can probably see from the number of posts I've made I'm a bit new here. I'm not a student and my background in math is a bit weak, what I need is a bit more than a tutor...I need a solution. Is that done here or is that against all the rules? I'm using the volume formula for a frustum pyramid and it gives me accurate answers (as it should)... :lol:
V= h/3(A1+A2+SqRt(A1*A2))
Where:
h=height
A1=Area of Large Base
A2=Area of Small Base

What I need is a formula that will give me B1 if I know the Volume, Height, and Slope...

Givens:
V= 1,007,436.00
h= 12.00
s= 3.00 {s= slope of 3:1 or 71.56505}

Solve for:
A1= 105,625.00 {large base}
b1= 325.00 {large base side} (A1^0.5) or sqrt(A1)
with an option for;
A2= 64,009.00 {small base}
b2= 253.00 {small base side} (A2^0.5) or sqrt(A2)

If it makes a difference when the formula is built...once I have solved for A1, I can take the sqrt of it to find the lenght of b1 sides. Then, I want to be able to make it rectangular by using a smaller width than the calculated side and specify the length by dividing A1 by the width, while keeping the same volume that was specified for the original calculation. This make sense to anyone or have I confused the entire issue...? I've been informed that this requires a quadratic formula which puts it way above my head.

I don't really need to know A2 and b2 but it would give me a way to cross check the validity of what I'm doing. I draw these as solids in AutoCAD so it's not a problem to check the volume once the object is drawn.

Any and all help you can give me on this solution will be greatly appreciated and I thank you for taking time out to help.
 
3ea2928d8ddedfbf5c8669ae6d3d2256.png


Isn't that the volume of a Frustum?

If so, try to reformulate the initial equation so that B1 is all by itself.
 
Euler said:
Isn't that the volume of a Frustum?

If so, try to reformulate the initial equation so that B1 is all by itself.

Yep, that's the formula for the volume...problem from there is, my math background isn't even close to reformulating this guy, to get where I want to be. BTW, if anyone cares to take a crack at this, the Volume, Height and Slope will be variables...the values I gave in the example are from a recent project that I just finished.
 
The closest thing I can get is:
eqn9349.png


You still have one B1 per side, but at least B1 is isolated…
 
Euler said:
The closest thing I can get is:

You still have one B1 per side, but at least B1 is isolated…

Thanks for the effort but I don't know the value of B2... I do know the slope of the sides, the height and the volume...that's all I will have to work with. I know this can be done, as my son-in-law put a crude formula together that works, just doesn't react the way I want and the formula is huge. He don't take to a lot of pestering so I thought I'd back off before I wear out my welcome.

There must be a way to plug in the slope (or degree of slope) instead of the B2 to make this work...? Just don't know how.
 
You are correct that one can solve for B1 explicitly, using the quadratic formula, but it requires great care. It is NOT pretty. However, this does not quite seem to be your question. With only the Height and the Slope you should be able to build a relationship between B1 and B2, THEN you will be closeer to your goal.
 
tkhunny said:
...but it requires great care. It is NOT pretty.

However, this does not quite seem to be your question.

With only the Height and the Slope you should be able to build a relationship between B1 and B2, THEN you will be closeer to your goal.

Ok, can you expand on the NOT pretty comment...? As for requiring great care I haven't a clue as to how to even get started.

I'm familar enough with right triangles that I can calculate the distance of the slope and the angles...I know the height and the angle of the slope so all the other sides can be calculated. Where exactly would that bring me in a relationship with B1 and B2...?
 
It was my suggestion that you don't want a clue about the "not pretty" part.

What you need, I think, is a relationship between B1 and B2, from the given information.

You have h = 12 and s = 3.

If you draw a cross-sectional view, just a trapezoid, marking verticals from the edges of the top to perpendiculars on the bottom, you see a rectangle, representing where the two bases are identical, a square the size of the top (B2) cut out of the bottom (B1). Also, there are right triangles on either end, representing where the bottom is larger than the top.

From s = 3 and h = 12, we see that the bottom extends beyond the top by h/s = 4, all the way around. Thus, if the side of the bottom is S1, the area of the top is B1 = S1<sup>2</sup>.

Further, the area of the top, B2 = (S1-2*(h/s))<sup>2</sup> = (sqrt(B1)-2*(h/s))<sup>2</sup>

Part of the volume formula can then be reduced to one less parameter.

Are we getting anywhere, yet?
 
tkhunny said:
It was my suggestion that you don't want a clue about the "not pretty" part.
I'm not so sure about that part...I'd like to know what you mean by the "not so pretty". Complex formula, long formula, frustrating formula, confusing formula, etc...?
What you need, I think, is a relationship between B1 and B2, from the given information.
We're both in sync on this point, it's my understanding that I can't get where I want to go until I can solve for B1 AND B2...correct?
a square the size of the top (B2) cut out of the bottom (B1). Also, there are right triangles on either end, representing where the bottom is larger than the top.
Sounds like I should be looking at some of the isosceles trapezoid formulas to better understand some of this...?
Thus, if the side of the bottom is S1, the area of the top is B1 = S1<sup>2</sup>. Further, the area of the top, B2 = (S1-2*(h/s))<sup>2</sup> = (sqrt(B1)-2*(h/s))<sup>2</sup>
Gota do some required field work but I will get back to digesting this when I'm finished in the field. Part of my confusion is that my model is upside down, i.e. the large base is always on the top...just a matter of me getting my head set that what we're working with here is rotated 180.
Are we getting anywhere, yet?
Yeah, some of the lights are starting to come on. Vaguely remember something being taught about the trapezoids some 45years ago but then again maybe I was sick that day. Soon as I get a grip on the above information I'll be pinging again. I like math, just don't understand it... :? Thanks for your efforts and nudges...a dim light is aglow over yonder, just a matter of me moving closer to increase the brightness.
 
Whoops! Revision.

I said:

"Thus, if the side of the bottom is S1, the area of the top is B1 = S1<sup>2</sup>"

I meant:

"Thus, if the side of the bottom is S1, the area of the BOTTOM is B1 = S1<sup>2</sup>"

Sorry for any resulting confusion.

It hadn't dawned on me that my frustrum would be upside down. While thinking on other possible assumptions I had made inappropriately, I wondered what your 's' really is. Is it the slant of a FACE or the slant of one of the corner EDGES? They are not the same thing. I assumed it was a face.
 
Well, OK, if you REALLY want to see it. These don't look particularly useful to me, but just to give you a flavor of their complexity:

First, without removing B2. It is quadratic in sqrt(B1) so you get two solutions. It is not immediately apparent to me that one or the other is positive.

http://img85.imageshack.us/img85/2831/b1withb20fu.gif

Second, after removing B2. This time, we introduced another quadratic level, making a quartic-type equation. This gives four solutions. Again, it was not immediately apparent to me which might be appropriate for a given set of data.

http://img46.imageshack.us/img46/8724/b ... ted5ii.gif

For what it's worth...
 
tkhunny said:
Well, OK, if you REALLY want to see it. These don't look particularly useful to me, but just to give you a flavor of their complexity: For what it's worth...
Oki, I understand what you mean by "not pretty", very similar to "downright ugly" and very complex and more than just a little bit confusing.
It is not immediately apparent to me that one or the other is positive.
A bit hard to tell which direction to go I'm sure.
This gives four solutions. Again, it was not immediately apparent to me which might be appropriate for a given set of data.
Even more choices than before, could wind up at the beginning before reaching the middle.
Thus, if the side of the bottom is S1, the area of the top is B1 = S12.
Thanks for posting the later explaination on this...I was looking at this when I mentioned the confusion of my model being rotated 180 from yours and thought my head wasn't connected. Makes more sense now.
It hadn't dawned on me that my frustrum would be upside down. While thinking on other possible assumptions I had made inappropriately, I wondered what your 's' really is.
What I have in the 'real world' is a hole in the ground...water retention pit that will be enclosed by an embankment (which involves another formula :roll: ). The goal is when they tell me how many acre feet of water they want to retain, I can quickly draw a pit with an embankment...building the embankment out of the material that came from the pit so it's all balanced out nice and tidy.

From s = 3 and h = 12, we see that the bottom extends beyond the top by h/s = 4, all the way around.
I've got a bit of time right now but have to go out again for most of the afternoon. I've hit my 1st stumbling block with the s=3 & h=12 and the h/s=4... This is a 3:1 slope meaning for every 3x there is going to be 1y. This makes the bottom extend 36 beyond the top...with h=12 and s=3 wouldn't it be (h)(s) resulting in x=36 and y=12...? Or haven't I gone far enough?
 
OK, you got me again. I went with an algebra definition of slope, not with a "roof" definition. These are just reciprocals.

This gives the difference of sides as 2*s*h, as you suggested.

New formulas for this version. They didn't get any better, but they shouldn't be TOO hard to program on a decent calculator.

http://img13.imageshack.us/img13/7240/b ... ope5dl.gif

Just for the record, the one that matches your solution is the third one. Oddly, the fourth one is the corresponding B2. I don't know why that would be. In any case, if you were able to program the formulas, you can simple check the results against your given volume. If you get the wrong volume, discard the solution. There are only four to try. Only one should work. The more I look at it, the more it appears there are only two good candidates. See those "+2" and "-2" in the four formulas, right about in the middle? It looks like the "+2" versions are candidates for B1 and the corresponding "-2" is a candidate for B2.
 
OK, I get it, now.

The formula for the volume is symmetric. See how B1 and B2 play the EXACT same role in the formula? If you switched them, you would get the same answer. This explains why the four answers contain two solutions for B1 and two solutions for B2. The "+2" versions being greater than the "-2" versions, the "+2" must be your B1. Still, only one should work.

We MAY be able to rule out the other one, but I haven't dicided how, just yet.
 
tkhunny said:
OK, you got me again. I went with an algebra definition of slope, not with a "roof" definition. These are just reciprocals.
Wasn't trying to get-ya and I certainly appreciate the responses. That's why I gave the angle with the 3:1 in the original post, thought this would be understood by the math people. Sorry... :oops:

New formulas for this version. They didn't get any better, but they shouldn't be TOO hard to program on a decent calculator.
I will program this right into AutoCAD so when I ask for a pit of xVolume it just draws it for me. Less user input the better... :D


The formula for the volume is symmetric. See how B1 and B2 play the EXACT same role in the formula? If you switched them, you would get the same answer. This explains why the four answers contain two solutions for B1 and two solutions for B2. The "+2" versions being greater than the "-2" versions, the "+2" must be your B1. Still, only one should work.
Oki, I'll have a go at these guys and see how they react. The nice thing about doing this in AutoCAD is that it will give me an instant answer as to the volume just by clicking on the drawing (it's a 3D solid). Gota go out for the rest of the afternoon but this will certainly give me something to play with if I have any 'free' time this long weekend.

THANKS for the support and the formulas, that was most kind of you. You have yourself a great weekend and I'll let you know which of the solutions works for what I'm doing.
 
tkhunny said:
OK, I get it, now.
Well I don't get it at all...? I haven't any way to post the image that I drew so will try to walk you through what I've done. I used the third formula down in your image...the one that begins with -1/3 and has the +2. I began with taking the square root of the guy in prens and multiplied this by the other formula within the same brackets. Then went to the left hand side and worked my way to the right. When that was done I divided by h and multiplied that by 1/3. I've written this like it would go into a spread sheet so it's not as clean as yours. Is this the right way to do this and should this give me a result for B1...? My final number is so large I don't even know how to read it but then I'm easily confused with anything greater than a billion. :?


Step1 = (((-3 * (h ^4)) * s ^2) + (9 * h * V)) ^0.5
Step2 = (6 * (h ^2) * s) + 2
Step3 = (Step1 * Step2)
Step4 = (-3 * V) – h
Step5 = (Step4 * Step3)
Step6 = s + ((4 * (h ^3)) * s ^2)
Step7 = (Step5 * Step6)
Step8 = (Step7 / h)
Step9 = (-1/3 * Step8)
 
Scrutch said:
Step1 = (((-3 * (h ^4)) * s ^2) + (9 * h * V)) ^0.5
Step2 = (6 * (h ^2) * s) + 2
Step3 = (Step1 * Step2)
Step4 = (-3 * V) – h
Step5 = (Step4 * Step3)
Step6 = s + ((4 * (h ^3)) * s ^2)
Step7 = (Step5 * Step6)
Step8 = (Step7 / h)
Step9 = (-1/3 * Step8)
Couple of misunderstanding or mistranslations, but only one variety.

a+b*(c+d) = a+(b*(c+d))
a+b*(c+d) is NOT = (a+b)*(c+d) <== Never do that again. :)

Try this:
Step1 = 2*[(((-3 * (h^4)) * s^2) + (9 * h * V)) ^0.5]
Step2 = (6 * (h^2) * s)
Step3 = h*(Step1 + Step2)*s
Step4 = (-3 * V)
Step5 = (Step4 - Step3)
Step6 = ((4 * (h^3)) * s^2)
Step7 = (Step5 + Step6)
Step8 = (Step7 / h)
Step9 = (-1/3 * Step8)

Word of Warning: I do not know that #3 on my list will always be the correct answer. You may have to take some steps to ensure you have the right one. For one thing, I think you need to do #3 AND #4 and see which is greater, unless someone is kind enough to demonstrate how we might be able to tell. I'm pretty sure #3 is greater, but I haven't actually sat down to work it out. As for #1 and #2, one may also need to calculate those and see that they simply give the wrong volume - then discard them as possibilities. So, you can get the right answer by chekcing in pairs. Try #1 or #2, if you get the wrong volume, then #3 or #4 is the correct answer. If you get the right volume, then #1 or #2 is the right answer. Once you know which pair has the right answer, pick the greater of the two. It's not trivial, but it can be done.
 
tkhunny said:
Couple of misunderstanding or mistranslations, but only one variety.
a+b*(c+d) = a+(b*(c+d))
a+b*(c+d) is NOT = (a+b)*(c+d) <== Never do that again. :)
I'm not sure how I did this at all, I know the difference of the above but I think it's the darn brackets that's confusing me as what to do and when to do it. :oops: Don't remember anything about brackets but I'm pretty sure that I remember that the syntax is * 1st, / 2nd, + 3rd and - 4th...but I may have the + and - backwards though, don't really remember that part.
Try this:
Step1 = 2*[(((-3 * (h^4)) * s^2) + (9 * h * V)) ^0.5]
Step2 = (6 * (h^2) * s)
Step3 = h*(Step1 + Step2)*s
Step4 = (-3 * V)
Step5 = (Step4 - Step3)
Step6 = ((4 * (h^3)) * s^2)
Step7 = (Step5 + Step6)
Step8 = (Step7 / h)
Step9 = (-1/3 * Step8)
Oki, this is way different that the way I was doing it. Thanks for helping me through the bracket thing...this brings the number down to 54+ million instead of the one that was two somethings above the trillion guy...? What I got for this using: V=1,007,436 h=12 s=3:
Step1 = 20,808
Step2 = 2,592
Step3 = 1,941,636,096
Step4 = (3,022,308)
Step5 = (1,944,658,404)
Step6 = 62,208
Step7 = (1,944,596,196)
Step8 = (162,049,683)
Step9 = (54,016,561)

I'll try the others with the syntax you have shown and see if I can get a more reasonable answer but looking at the above result I may still be doing something wrong. Negative 54million seems a bit high. Using 1,007436 for the volume the sides of the large end shoud be 325' or 3900" per side and the small end should be 253' or 3036" per side given a 3:1 slope and 12' depth.
It's not trivial, but it can be done.
I hear you there... :shock: I would like to thank you for the help you're providing, I really appreciate it.
 
Step3 = 1,941,636,096
Shhheeezzzz...did it up royal...used a * instead of a +... :oops: I've dropped this number down to 842,400 but the final answer still doesn't make sense...yet. Off to triple check my calcs before I start running off at the mouth. :oops:
 
tkhunny said:
For one thing, I think you need to do #3 AND #4 and see which is greater, unless someone is kind enough to demonstrate how we might be able to tell. I'm pretty sure #3 is greater, but I haven't actually sat down to work it out.
OK, by entering the formula the way you show in the steps the final answer is (105,625). If I take the square root of this guy I get 325, which is the number I'm looking for. This make sense to you...?
Also, is there a reason the final answer would be negative rather than positive or does that mean I need to use the -2 guy instead of the +2. I haven't tried them out yet but I'm headed there now. This is going to be Great! :D
 
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