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?
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
and the area of the base is just the regular formula for the area of a circle
so that means the volume is
If you want to measure it in terms of the diameter instead, you have
so for the volume, you get
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.
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:
Expanding, simplifying, and solving the resulting linear equation for R gives
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)