# Volume of a Cylinder

A cylinder is a three-dimensional shape circular in cross section. Cylinders are very common, from cans to tubes, to internal combustion engines. This instructable will show you how to calculate the volume of a cylinder.

What you may not know is that you probably already know how to do this; the instructable will extend the instruction to show how calculating the volume of a cylinder is similar to calculating the volume of other shapes.

This instructable is part of the Burning Questions Round 6.5 contest... If you like this, please vote for me!
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## Step 1: Gather equipment

You'll need to be able to measure the cylinder. A digital caliper is a good purchase if you're going to be measuring a lot, and there are adequate units for under \$30. If you need to measure internal dimensions, then a caliper is the way to go. You'll need:

1. Some type of measuring device
2. A calculator (some units have a dedicated pi key, which would be handy)
Amnesia Wes says: Aug 6, 2009. 7:09 PM
(removed by author or community request)
Amnesia Wes in reply to Amnesia WesAug 7, 2009. 4:55 PM
Without doing all the calculations: take a tub, fill it to the top with water. Put your cylinder into the water. Take the amount of water that overflows from the tub; pour it into a measuring cup. The measuring cup will then tell you exactly how much volume the cylinder is. Archimedes did this eons ago, screaming Eureka! remember? One of the things I forgot to mention is that this method can be used to determine the volume of any kind of object, whether it's a ball, a cylinder, a cone, etc. In fact, it can even be used to determine the volume of irregular shapes, for example: ask a mathematician to determine the volume of a bowling pin. It'll take him quite some time because of all the curves and dimensions, whereas with this method it will only take mere minutes.
Cooldeal in reply to Amnesia WesJul 10, 2011. 7:39 AM
What about the thickness of the walls of the cylinder? What if the walls were only thick on some of the sides?
KEUrban (author) in reply to CooldealJul 10, 2011. 9:24 AM
This instructable assumes that the walls are infinitely thin, but to answer your question, just measure the inside diameter of the cylinder and then the wall thickness is immaterial.

However, if you measure the outside diameter of a cylinder and calculate the volume, then measure the inside diameter and calculate THAT volume and finally subtract the smaller of the two volumes from the larger you'll be left with the volume of the materials in the walls of the cylinder.

Make sense?
Cooldeal in reply to KEUrbanJul 10, 2011. 6:05 PM
No. Just charge me for the beer I drink out of it instead of the whole volume. Too much computing for me to handle. But I can always fill it then measure what comes out of it.
ariberman says: Feb 25, 2009. 7:49 AM
Good overview. Now, can you show me how to do partial volume calculations in a horizontal cylinder? This seems a lot more tricky.
KEUrban (author) in reply to aribermanFeb 26, 2009. 4:42 AM
It is a lot more tricky. What you're asking is how to calculate the area of a "circular segment." If you lay a cylinder on its side and fill it partially, the flat surface created by the material in the cylinder describes what's called a chord. The typical way to calculate the area of a circular segment requires that you know the angle between the two lines of radius where the chord intersects the perimeter of the cylinder. See this web site for a computation: http://mathworld.wolfram.com/CircularSegment.html

I am working on an Instructable that approximates the area from the depth of the substance filling the cylinder.

In any case, once you know the area of the filled portion of the cylinder, the volume is found simply by multiplying the area by the length of the cylinder, like I describe in step 4 of this instructable.
espdp2 in reply to KEUrbanJun 20, 2011. 9:06 AM
This can finally lead me to the answer to a real puzzler that was posed on Car Talk on NPR a few months back! A caller had a big-rig with the horizontal (approximately) cylindrical fuel tanks. His gas gauge was on the fritz, and he wanted to know how to tell how much fuel he had. 1/2 tank is easy enough. 1/4 tank or 10% had Ray and Tom stumped, and they put out a request for their listeners to solve it.

Assume a 20" diameter tank, perhaps 40" long. Let's leave a little gap at the top for fuel expansion and such. Where do you put the mark on a dipstick for a quarter tank, in inches?

I hated math, until I started running into real-world applications like this, and now I wish I'd paid a lot more attention! BTW, my dad is a high school math teacher, and he was stumped too. Thanks! :-)
KEUrban (author) in reply to espdp2Jun 22, 2011. 4:52 AM
I still have the 'ible in progress to approximate the value, but I'm just too busy to pull it together.

Search for something called "Gordon's Approximation" to estimate the area of a circular segment, which you would then multiply by the length to generate volume.

Thanks for your comment.
espdp2 in reply to KEUrbanJun 23, 2011. 12:54 PM
I'm trying to solve the "Quarter Tank Problem," where segment area is known, and I need to solve for length, "L", which is where the notch on the dipstick should go.
espdp2 in reply to espdp2Jun 20, 2011. 9:37 AM
Yes, I should have paid more attention indeed. See here:

I'm trying to find "H" height. I've forgotten my math formula magic. :-/
unigamer says: Jan 30, 2009. 10:31 AM
pi is irrational and is therefore by definition infinite. A clear instructable though.
KEUrban (author) in reply to unigamerJan 30, 2009. 3:39 PM
Thanks for the comment! Of course you're absolutely correct. I didn't think adding in another mathematical construct like irrational numbers added anything from an instructional standpoint. My philosophy can be summed up by something I read once: "Why measure with a micrometer when you're going to mark it with chalk and cut it with an axe?"
espdp2 in reply to KEUrbanJun 20, 2011. 8:56 AM
Love that quote! Sometimes "good enough" is good enough.
uguy says: Jan 30, 2009. 7:56 AM
Pie are round.
Poppa Chubby in reply to uguyDec 5, 2009. 11:46 PM