## Introduction: Volume of a Cylinder

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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!

## 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)

## Step 2: Master a Few Terms

The base of a cylinder is a circle. There are three critical dimensions of a circle and one mathematical construct you need to know:

Diameter: The diameter is the measurement across the widest dimension of the circle. Any measurement of the diameter goes through the exact center of the circle. If the measuring device doesn't go through the center of the circle, the measurement will be something less than the diameter of the circle.

Radius: The radius of a circle is exactly half of the diameter.

Circumference: The circumference is the measurement around the edge of the circle.

The fourth term you need to know is a mathematical construct called pi. Pi is represented in mathematical expressions using the Greek letter pi (see the images below). Pi is the number that represents the ratio of the circumference of the circle and its diameter. In English, that means if you measure the circumference of a circle and divide by the diameter of the same circle, the result will always start out 3.14159... Why the three dots after the numbers? It's because pi continues to infinity (at least as far as we know!) without repeating.

## Step 3: Calculating the Volume of the Cylinder

1. Measure the diameter of the cylinder, and divide it by two. This is r, the radius.

2. Measure the height of the cylinder. Make sure you use the same units of measure that you used when you measured the diameter. In other words, if you measured the diameter in inches, measure the height in inches too.

3. The superscript in the formula tells you to square the radius, which means to multiply it by itself.

4. Take the square of the radius and multiply that number by pi (use 3.14159).

5. You now have the area of the circle that makes up the cylinder base. Area is expressed as square units; because we measured using inches the area of our cylinder base is square inches.

6. Multiply the area of the cylinder base (the circle) by the height of the cylinder.

7. You now have the volume of the cylinder in the same units you used to measure the parts. Volume is expressed as cubic units; because we measured using inches the volume of our cylinder is cubic inches.

## Step 4: Extending What You Learned

You probably have already done something like this before, just not with a cylinder. Think about the volume of a box. How do you calculate that? You multiply the length of the box by the width, then multiply that number by the height to calculate the volume.

Look at the formula for calculating the area of a circle below. Since we know that squaring a number means multiplying it by itself, we could rewrite the formula as pi x r x r. As it turns out, calculating the area of a circle is a special case of calculating the area of a squashed circle, known technically as an ellipse. An ellipse has two measurements, one vertically oriented through the center of the ellipse called the minor axis (to calculate the area we need 1/2 of the minor axis length, which we'll call a), and one through the center on the long axis of the ellipse called the major axis (let's call half the length of the major axis b). To calculate the area of an ellipse, the formula is pi x a x b. Multiply the area of the ellipse by the height, and you have the volume of a tube with an elliptical cross section.

Thus, if you can measure the area of the end of tube or bar, you can measure the volume once you have the length.

## Step 5: What Can You Do With This?

Suppose you have a cylinder with an internal volume of 100 cubic inches. How much would it weigh if you filled it with water?

The US Geological Survey says water weighs around 62 pounds per cubic foot (62.416 pounds per cubic foot at 32 degrees Fahrenheit, and 61.998 pounds per cubic foot at 100 degrees Fahrenheit). A cubic foot is a volume 12 inches wide, 12 inches long, and 12 inches high, or 1,728 cubic inches. Thus one cubic inch of water weighs 0.036 pounds. Therefore our cylinder would hold about 3.6 pounds of water (100 cubic inches times 0.036 pounds per cubic inch).

Below is a list of formulae to calculate the area of various plane figures; just multiply by the length of a tube to determine the volume of a cylinder or tube!

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## 20 Comments

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.

What about the thickness of the walls of the cylinder? What if the walls were only thick on some of the sides?

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?

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.

Good overview. Now, can you show me how to do partial volume calculations in a horizontal cylinder? This seems a lot more tricky.

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.

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! :-)

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.

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.

Yes, I should have paid more attention indeed. See here:

http://www.mathopenref.com/arcradius.html

I'm trying to find "H" height. I've forgotten my math formula magic. :-/