Introduction: 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!
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!