Cylinders

A can of soda. A silo with wheat. A tanker with diesel fuel. You encounter the cylinder shape everywhere, and sometimes you need to know how much content it can hold. If you understand how to calculate the volume of a prism and the area of a circle, this won't be a problem.

If not, it might be a good idea to take a look at those lessons first. The cylinder is a straight pipe, circular at the ends and with an arched wall that binds the two circles together. To get the cylinder's volume, you calculate the area of the base - the area of one of the end circles, and multiply it by the cylinder's height. The area of the base times the height equals volume. And the area of the base is calculated like any other circle's area: Pi times radius squared.

This is how much a cylinder can hold, or its volume. But let's say you want to make a cylinder and you need to know how much material is needed. Then you need to calculate a surface area. The area of a circle either at the bottom or at the top is easy to calculate. A circle's area is pi times radius squared.

But how do you calculate the area of the arched surface? Imagine that you cut the cylinder open lengthwise and flatten the side surface, like this. You get a rectangle. The rectangle's height is the cylinder’s height, so we know that. But what about the rectangle's base?

Where does that come from? Well, before, when it was a cylinder, the base used to be the circle at the bottom. So, the rectangle's base is equal to the circle's circumference. The base is therefore: Pi times diameter. Or Pi times radius times two, if you prefer that.

And here's your answer. The surface area of the side of a cylinder equals: the circle's diameter times pi, times the cylinder's height. So, here they are, the formulas for the cylinder's volume and the area of the side surface. Pause and write them down. You'll need them when you do the quiz!