Formula for Calculating the Gallonage of a Circular Tank

Is there anyone out there who can tell me how to figure the gallonage of a Large Circular Water Tank. I know the total gallonage size of the tank but I need to know how many gallons are in the tank when it is 1/4 and 1/3 full and so on. I am not a Math wizard, so this may sound elementary to some people. Any help would be appreciated. You can post here or email me.
Thanks, Tim

Tim,
It depends on the shape of the tank, and what you mean by 1/3, 1/4 etc full. Is the tank cylindrical or does its radius change? Also, by 1/3 full 1/4 do you mean 1/3 or 1/4 the volume of the tank, or do you mean water is up to 1/3 / 1/4 the height of the tank?

If the tank is cylindrical and you mean 1/3, 1/4 the volume, you can just multiply the tank capacity by 1/3 or 1/4 or whatever.

If the tank is cylindrical (and its circular radius doesn't change) and it is lying on its side, and you mean 1/3, 1/4 the way up the tank, you have to use a definate integral.

Let me know what you mean. If you need the formula for the definate integral i would be glad to help you

Tim, if you need the formula, you can actually do it without calculus. Go here and it will explain it (at bottom of page). You should just be able to plug in variables and go:

http://mathworld.wolfram.com/CylindricalSegment.html

Thanks Nick, I should have made my request more specific. The tank I am referring to is a large water tank and it is laying on its side. Its dimentions are 12 feet in Diameter by 40 feet in length and is supposed to hold 40,000 gallons. What I need to know is how many gallons there would be in it, for instance, if there was 12 inches of water in it. The only way I have of measureing the water is a graduated stick that tells me how many inches there are from the bottom. I was wondering if there was any kind of formula for figuring how many gallons there would be at any given level.
Thanks, Tim

[convert-me.com]

Tim,

First of all, if your measurements are exact (40 feet long, 12 feet in diameter), the volume of your tank is a bit less than 40,000 gallons:
V = 40 * 36 * Pi = 4524 cubic feet = 33840 gallons.

Below is the folmula to calculate the amount of water in your tank. Not an easy one. :)

Let H is the level of water from the bottom that you have measured. The volume of water in the tank in cubic feet can be calculated as follows:

V = 40 * [ Pi*18 + (h-6)*sqrt(12*H - H2) + 36*arcsin(H/6-1) ]

You will need an enginering calculator capable of doing arcsin function. To convert from cubic feet to gallons, you'll have to multiply the result by 7.48.

Or you can use this chart for your convenience:

Code:

  H (feet)  |   V (gallons)
-----------------------------
   0,0      |   0
   0,5      |   482
   1,0      |   1347
   1,5      |   2441
   2,0      |   3707
   2,5      |   5107
   3,0      |   6616
   3,5      |   8210
   4,0      |   9874
   4,5      |   11590
   5,0      |   13346
   5,5      |   15126
   6,0      |   16919
   6,5      |   18712
   7,0      |   20493
   7,5      |   22248
   8,0      |   23965
   8,5      |   25628
   9,0      |   27223
   9,5      |   28731
  10,0      |   30132
  10,5      |   31397
  11,0      |   32492
  11,5      |   33356
  12,0      |   33840
---------------------------


By the way, the calculations assume that your tank is placed horizontally on a flat surface. If it is not so, your figures will be different.

Last edited by convert-me.com on Tue Jul 20, 2004 9:30 am, edited 1 time in total.

Thanks for the Great Chart, that is just what I was looking for and thanks for the information on the size of the tank, I was told it was 40,000 gallons, so I just assumed. This is a wonderful place to get information, I will recommend it to my friends.Again many thanks.
Tim

Hi again, I am sorry but I made a mistake about the size of the tank I needed the information for. The dimensions of the tank are 50 feet by 12 feet instead of 40 feet. The chart you provided was perfect for me. If you could recalculate it again, I would appreciate it. I do not have the expertise in Math to figure it out and I do not have the proper calculator. I did plug in the numbers to figure the Cubic Feet of the tank but now a chart like the one you provided before would be perfect. Sorry to have not had the right dimensions for the tank in the first place. Thanks again for your help.
Tim

No problem Tim.

Now it looks your tank holds even more than 40,000 gallons - 42,300 to be exact. :)

By the way, the dependency on the length of the tank is linear, so to get the new chart you just need to multiply the values in the old one by 5/4. Here is the new one:

Code:

  H (feet)  |   V (gallons)
-----------------------------
   0,0      |   0
   0,5      |   603
   1,0      |   1684
   1,5      |   3052
   2,0      |   4634
   2,5      |   6384
   3,0      |   8269
   3,5      |   10263
   4,0      |   12342
   4,5      |   14488
   5,0      |   16682
   5,5      |   18908
   6,0      |   21149
   6,5      |   23391
   7,0      |   25616
   7,5      |   27810
   8,0      |   29956
   8,5      |   32036
   9,0      |   34029
   9,5      |   35914
  10,0      |   37665
  10,5      |   39247
  11,0      |   40615
  11,5      |   41695
  12,0      |   42298
------------------------


Regards,
Sergey

Thanks Sergey, That will help me a lot. I wish I was a Math expert, but sadly that is not my long suit. Thanks again.
Tim

Tim,
It depends on the shape of the tank, and what you mean by 1/3, 1/4 etc full. Is the tank cylindrical or does its radius change? Also, by 1/3 full 1/4 do you mean 1/3 or 1/4 the volume of the tank, or do you mean water is up to 1/3 / 1/4 the height of the tank?

If the tank is cylindrical and you mean 1/3, 1/4 the volume, you can just multiply the tank capacity by 1/3 or 1/4 or whatever.

If the tank is cylindrical (and its circular radius doesn't change) and it is lying on its side, and you mean 1/3, 1/4 the way up the tank, you have to use a definate integral.

Let me know what you mean. If you need the formula for the definate integral i would be glad to help you

Tim,

First of all, if your measurements are exact (40 feet long, 12 feet in diameter), the volume of your tank is a bit less than 40,000 gallons:
V = 40 * 36 * Pi = 4524 cubic feet = 33840 gallons.

Below is the folmula to calculate the amount of water in your tank. Not an easy one. :)

Let H is the level of water from the bottom that you have measured. The volume of water in the tank in cubic feet can be calculated as follows:

V = 40 * [ Pi*18 + (h-6)*sqrt(12*H - H2) + 36*arcsin(H/6-1) ]

You will need an enginering calculator capable of doing arcsin function. To convert from cubic feet to gallons, you'll have to multiply the result by 7.48.

Or you can use this chart for your convenience:

Code:

  H (feet)  |   V (gallons)
-----------------------------
   0,0      |   0
   0,5      |   482
   1,0      |   1347
   1,5      |   2441
   2,0      |   3707
   2,5      |   5107
   3,0      |   6616
   3,5      |   8210
   4,0      |   9874
   4,5      |   11590
   5,0      |   13346
   5,5      |   15126
   6,0      |   16919
   6,5      |   18712
   7,0      |   20493
   7,5      |   22248
   8,0      |   23965
   8,5      |   25628
   9,0      |   27223
   9,5      |   28731
  10,0      |   30132
  10,5      |   31397
  11,0      |   32492
  11,5      |   33356
  12,0      |   33840
---------------------------


By the way, the calculations assume that your tank is placed horizontally on a flat surface. If it is not so, your figures will be different.