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I have a trough which is a circular container. How do I determine how many gallons of water it takes to fill up the trough? I was thinking that we measure the height and the width but I think it's a bit different with a cylinder. What's the formula?

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The volume of a right cylinder is given by $V=\pi r^2 h$, where $r$ is the radius of the base/top and $h$ is the height/length of the cylinder. –  Arturo Magidin Jun 23 '11 at 19:33

3 Answers 3

Your question makes it sound like the trough is an upright cylinder, with its base flat to the ground. If that's not the case let me know, and I will edit this answer. If it is the case, then the formula for the volume of a cylinder is

Volume = Area of base $\times$ height

and the area of the base is just the regular formula for the area of a circle

Area = $\pi \times \text{radius}^2$

so that means the volume is

Volume = $\pi\times \text{radius}^2 \times \text{height}$

If you want to measure it in terms of the diameter instead, you have

Diameter = radius / 2

so for the volume, you get

Volume = $\pi \times \text{diameter}^2 \times \text{height} / 4$

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thank you. That was helpful. I ask because I want to add chlorine to the trough enough to disinfect. How do I convert volume to gallon? –  netrox Jun 23 '11 at 19:57
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1 cubic foot = 7.4805 gallons, so multiply the number of cubic feet by 7.4805 to get the number of gallons. If you are measuring in inches, 1 cubic foot = 1728 cubic inches, so 1728 cubic inches = 7.4805 gallons; multiply the number of cubic inches by 7.4805 and then divide by 1728 to get the number of gallons. –  Arturo Magidin Jun 23 '11 at 20:45
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@Arturo thanks for the edits! –  Chris Taylor Jun 24 '11 at 8:03

If the trough is horizontal and you want to partially fill it, you could look at Wikipedia on circular segment. Multiply the area from there by the length of your trough.

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If the trough is horizontal and assuming the depth is less than the radius of the cylinder, you can calculate the area of a right angle cross section with a little trig or algebra or both.

Let W = the width, L = the length, and D = the depth, and R = the unknown cylinder radius.

Then the radius of the cylinder can be calulated by solving the equation:

$R^{2}=(R-D)^{2}+(\frac{W}{2})^{2}$

Expanding, simplifying, and solving the resulting linear equation for R gives

$R= \frac{W^{2}+4D^{2}}{8D}$

Now some simple trig allows you to calculate the portion of a complete circle's area that the cross section holds, and that multiplied by the length will give you the trough's volume. You'll need to convert that to gallons by looking up the conversion factor (one cubic foot of water is just under 8 gallons)

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