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# Volume by Average End Area

This Content Originally Published by a land surveyor to Land Surveyors United Network

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### Replies to This Discussion

Rather than Average End Area try this modification using the geometric mean.

V = L/3(A1 + SQRT(A1*A2) + A2)

L is distance3 between sections at stations 1 and 2.

A1 is the cross-sectional area at station 1.

A2 is the cross-sectional area at station 2.

Gives the correct volume for a pyramidal shape.

Hi Barry, thanks for the reply, Great equation, I know i seen it somewhere before, but thanks for sharing it. A1 and A2 are still unknown, The purpose of what i am showing is for students,who use a RPN hand programmable calculator, to use on their PE or PLS Exams.Only certain calculators can be used. My other purpose was to show how it was done years ago using these programs, before pre- program devices came out. In these test you must program or use key stroke programs. Nothing else is allowed. This is also to show how it was done before computers. And how Average by End Area was used. Can you show me this in RPN, that would be great.

G (a1,a2,a3) = (a1 a2 a3 ) 1/3

I would have to write yours as

V = L / 3 ( A + √ ( A * B ) + B)    ,in my HP 35 that is the calculator in this discussion, It can be used on the test.

I also seen a program for this , i will have to look, but that is why i used Average by end Area.

Thanks  on any math or programs.

But this is how my HP 35s Program works to do X-Sections.So lets begin. Sta 1 is 0+00, the next Sta 2 is 0+25. your Center line is the Baseline that equals 0+00 right and left, to the right is positive + to the left - is negative.And that is how the program knows what is left and what is right from the CL center line of the Road Bed,or center line of the cut.You will Enter the Elevations and key in the Distance and R/S on a beginning point. What point it does not matter,but you have to end on this point,clockwise or counterclockwise, but whatever it must be done the same thru the program.
I used the center line,As a starting point,as a Rule, easy to start and end and easy to remember.And this is the place where the X-Sections starts. Just like you would use a planimeter. so Sta 1,0+00 Begins the cuts your second Sta 2 , could be at 0+01 if you like, This makes the area of 0+01 and the interval of 1, 1 x the Area ,but this example starts at Sta 2@0+25 and it is the same as Sta 1@0+00, so you will be multiplying by 25 times Sta 2 of 0+25, so this area will be between 0+00 to 0+25, using 0+01 can be done if there is a big difference in the stations template Elevations, or natural ground so the first X-Section is @STA 2,0+25. starting at 0+00 CL @ 0+25 we are going RIGHT, in this example, Using the elevation of 0.0,to show the program can use negative values,both in Elev. and Dist.

V 001  is the routine.
V 021 is a calling subroutine for interval input.
V 003 is a nested subroutine, it can not be called.

0+00 CL Rt. + / Lt. -
STA 1 is 0+00,   Note: First sta has 0 intervals
XEQ V 021, display = 0.0000 ,

STA 2 IS 25+00
XEQ V ,ENTER, V001
(Road Sta.2,0+25 ,and 0+00 CL Rt.+/Lt.-)

START HERE

Volume by Average End Area
CounterClockWise Motion  to the Rt.
( as above )
XEQ V 021, display 0.0000, STARTS PROGRAM

XEQ V ENTER, V001,run

0.0 ENTER 0.0 R/S ..........AS ELEV. ENTER, DIST,R/S

0.0 ENTER 10.0 R/S.........AS """"""""""""""""""""

2.0 CHS+/- ENTER 12.0 R/S... AS -2 ENTER 12 R/S

7.0 ENTER 20.0 R/S......... AS ELEV. ENTER, DIST,R/S

6.0 ENTER 3.0 +/- R/S..... AS 6 ENTER,-3 R/S

7.0 ENTER 18.0+/- R/S.... AS 7 ENTER,-18 R/S

2.0+/- ENTER 12.0+/- R/S... AS -2 ENTER-12 R/S

0.0 ENTER 10.0+/- R/S........ AS 0 ENTER-10 R/S

0.0 ENTER 0.0 R/S.....AS 0 ENTER 0 R/S....... Back at starting point

X-section complete,Now you want to calculate this section
and 25 ft . back to 0+00 Sta1 for the area in that 25 ft. so

Key in 25,
XEQ V021 ENTER...... interval subroutine of V001

RCL E =100.0000.......................Int. Vol.(cu yds.)
RCL D=216.0000........................Area (sq.ft.)

Initialize for next Sta.

[ R/S ]

( RD.STA.3,0+75,AND 0+00 CL Rt.+/Lt.-)

0.0 ENTER 0.0 R/S..........ELEV. ENTER,DIST,R/S

0.0 ENTER12.0 R/S.........""""""""""""""""""""

1.0+/- ENTER 14.0 R/S....-1 ENTER 14 R/S

1.0+/- ENTER 15.0 R/S ....AS -1 ENTER 15 R/S

10.0 ENTER 30.0 R/S.......ELEV. ENTER,DIST,R/S

8.0 ENTER 6.0 R/S...........""""""""""""""""""""

7.0 ENTER 21.0+/- R/S.....7 ENTER -21 R/S

4.0 ENTER 17.0+/- R/S......AS 4 ENTER -17 R/S

1.0+/- ENTER 10.0 +/- R/S ...AS -1 ENTER -10 R/S

0.0 ENTER 8.0+/- R/S..........AS 0 ENTER -8 R/S

0.0 ENTER 0.0 R/S......... AS 0 ENTER 0 R/S....... Back at starting point

X-section complete,Now you want to calculate this section
and 50 ft. back to 0+25 STA 2 for that area in that 50 ft.
so key in 50,
XEQ V021..........interval subroutine of V001

RCL  E=497.6852.........................Int.  Vol. ( cu.yds.)
RCL D= 321.5000............................Area (sq. ft.)

As you see it can be done, Little or Big projects. no it's not as fast
as a computer program. but your doing it ,this way,because you don't have one.
And you can not take them to a test. This can be used on any X-Section
System for  Volume by Average End Area.
And that's how it is done on a HP 35s calculator.
This is my Program,I have sole rights to it.
This program was written by Billy Brooks,
prior to this Date of 09-18-2016 .
for the HP 32s,33s and 35s.
all three have different programs.

Volume by Average End Area, HP 35s

1.LBL V...................21.RCL D
2.CLx..................... 22.RCL C
3.STOP................... 23.ABS
4.STO x A................24.STO D
5.RCL A...................25.  +
6.STO - B.................26. 54
7.R ↕ dn...................27.÷
8.X<>Y.....................28. x
9.STO x Z.................29.STO E
10.RCL Z..................30.STO + F
11.STO + B...............31.RCL F
12.R↕ dn...................32.STOP
13.STO A..................33. CLx
14.R↕ dn...................34.STO C
15.STO Z..................35.STO B
16.RCL B..................36.STO A
17. 2.........................37.STO Z
18. ÷.........................38.GTO V003
19.STO C..................39.RTN
20.GTO V003

As you can see this program is simple and short, so it can be programed Quick.
And is simple to use, I will later add labels and instruction line in the program lines.
To make it a little more complex and more user friendly.
But for now it serves it purpose

Today in Modern surveying and Engineering, large computer programs took over all this math and calculations. But just think how much work was put into just a small project for these Volume Average End  Area's. That does not include all the math of running levels ,them being reduced  and the X- section plotted on X-section paper.  But today's students have to take on this task of knowing how to do this and others like it. That is called Foundation.

First the Formulas i used:

Vol =( AREA í + AREA 1 - 1 ) INT / 2

AREA = 1/2 [ EL 1 ( D2  - Dn ) +... + ELn ( D1 - Dn - 1 ) ]

where : VOL =Average volume between two stations .

............AREA =Cross-sectional area at a station.

............INT =Interval between stations.

............EL = Elevation at a point on a cross- section.

............D =Horizontal distance ( offset ) from centerline.

............i =Subscript referring to current point.

............n = Subscript referring to the last point

............Numeric subscript refers to point or station number

Note: Volumes are calculated in cubic yards, areas in square feet.

If you like to have volumes calculated in cubic feet, then do this.

Delete  54 on line 26 and replace it with 2 in it place.

Hope all this helps.

Thanks again Survenator for the FAV., I can always count on you.

billy

Yes, it is called foundation. Most area formulas were derived from calculus which was invented by Sir Issac Newton. My HP 50G has derivative & integral libraries in it. I would have to break out the text to evaluate anything but a basic integral today but I will always remember the principles of how they work. The real advantage of the computer is that it allows us to do a better job because we can be much more analytical & do calculations that were not done before because they were too labor intensive.
Was it Billy that hit on units? Since I admit to being a dinosaur I should be right at home with the units we insist on using. Why do we insist on using the odd ball collection of units we call the English system? The foot is as good a unit as any but the units we associate with it do not constitute a system. The only reason to use meters is that it is a true system & that is what the rest of the world uses.

Hi Charlie, This Topic, is about how it was done,and how students or surveyors today who are taking their test will have to do it today on a Exam . There is no option for using a Computer. these are not computerized Exam.  You can not also have a pre-program card, or module or a downloaded program in memory.  In fact you must be a key stoke program Calculator. This program is to help them Before not After. No matter what anyone's opinion is about using a computer, it does not apply here. I am not trying to promote this as a everyday solution. I use a computer everyday and write programs in them. This is to help those who must do it on a calculator, they have no other choice. This is not what is Best to use. This is what they have to use. And how does Units fit into all of this, I do not recall saying anything about Units, and what is the meaning of  this statement.  Was it Billy that hit on units?. just what exactly are you saying. I would like to know. I only showed this in the Units we used. meters are no problem. but i do not know if that is the issue here so please clarify yourself.

http://ncees.org/exams/calculator/

Hey Billy,
My apologizes. My wife keeps telling me I have an attention deficit disorder & maybe she is right. Seems like I got my wires crossed. I was following the thread on area calculations & failed to notice the post I was replying to was in regard to road volume calculations using average end areas. Since the discussion was still area, I failed to notice my error. I have a bad habit of wandering off topic but not that bad.
Yes, I know you can't use a programmable calculator for the surveyor's exam but I wasn't aware the discussion was restricted to calculations for the exam. My main focus is the field.
Units? O.K., you got me again. I was getting a bit off topic but what sparked that was when you described how to modify your formula to change units. Units are a pet peeve of mine & it doesn't take much to get me going. The collection of odd ball units we use requires constant conversions. The worst example is feet & inches which surveyors don't generally use for obvious reasons. I call that BAUs for bastard architect units. The units commonly used in road construction are not much better. The lengths are measured in feet. The cut & fill is measured in cubic yards. Drainage is measured in gallons per minute or cubic feet per second. I think that's right. I haven't designed a road in a long time. I once designed & surveyed roads for the US Forest Service among other entities. Superelevation is feet per foot & I think surface runoff is inches per second (it's been a long time).
O.K., I'll get off my soap box & promise to restrict my comments to the topic in the future.

No need to apoligise.

Didn't see a thing wrong with your replies Charlie.

Volume by Average End Area is the topic heading - nothing else.

well maybe i just stop using them and put a Big ?????? marks. then everyone can guest what the discussion is about, That's is why books have titles, But i forgot you can not judge a book by it's Cover, well i guess that means the title's too.

Have a great day, and Barry thanks for the equation, but i am still waiting on the RPN program to it. That would be great ,like i said.

O.K., I give up. What does that mean? Stop using what? I don't believe I ever suggested that we stop using anything. I have already apologized for misinterpreting the two posts. I didn't mean to offend anybody.

Dear Charlie,  That reply was not to you, all is well between me and you. your friend ship means a lot to me. Why do we have topic headings, are they just for looks are to get someone attention. This site is to share and help your fellow surveyors,  that's is why it's Title  has United in it. We at times may have a different opinions, but to disrespect someones opinion or to put them or their idea down,this is not in the cards. I respect everyone here and in return I expect the same. The best thing is everyone try to get alone.  If someone does not want to be a friend with someone else ,that is your right, but just keep it to yourself and everything will be find. To you Charlie I hope this explains that I was not talking to you. And there is never nothing wrong with a apology, even when it is not needed and if i have ever done anything wrong to you I apologize for it also. You are a Great contributor to this community.

Best Regards

billy

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