Cylinders and Cheerios
Have you ever thought about why grain silos are cylinders?
What would happen if they were taller, skinnier, fatter?
Would they hold more grain?
We are going to show how two different cylinder shapes hold a different amount of Cheerios.
- Two clean sheets of typing or notebook paper (8 ½ inches by 11 inches) Note: They could be any rectangles of the same size.
- A box of Cheerios (you can eat the ones we use after the experiment) (or a bag of popcorn or a quart of sand)
- A big bowl
- Tape the two sheets of paper into cylinders, one with the long sides together, one with the short sides together.
- Do not overlap the ends of the paper. Line up the ends and tape them together.
- Put the tall, skinny cylinder inside the short, fat cylinder
- Put the nested cylinders into a bowl
- Fill the tall, skinny inside cylinder with Cheerios
- Pull up the tall, skinny cylinder, letting the Cheerios fall into the short, fat cylinder
What Should Happen?
The Cheerios that filled up the tall, skinny cylinder will not fill up the short, fat one.
You used the same size paper to make both cylinders. This is called the lateral surface area.
But, that’s not what determines the volume of a space.
For the cylinder, the volume is the area of the circle at one end times the height of the cylinder.
As the tall, skinny cylinder easily fits inside the short, fat cylinder, you can see that the end of the short, fat cylinder is larger than the tall, skinny cylinder.
Even though the skinny cylinder is taller, it’s not enough to make up for the fact that the short cylinder is wider around.
Two calculations prove it, as you figure out the volume of both cylinders.
The first calculation is the area of a circle, or the end of the cylinder.
It is calculated by multiplying the radius (half the distance across the widest part of the circle) by the radius (r2).
Then multiplying that number by pi (3.14), giving you the area of a circle = 3.14 * r2
The second calculation is multiplying the area of the circle by the height of the cylinder.
Area * height = volume
You have squared the radius while only multiplying by the height.
This is why the radius, or diameter of the circle, is what most affects the volume.
Even though you used the same sheet of paper in different shapes.
The surface area inside the cylinders is the same. You used the same size sheet of paper, after all.
But, the air the paper surrounds is a different volume.
The shape you choose changes the amount of air it can contain.
When Might This Matter?
If you were a farmer, you might very much care about building a silo with the largest volume to hold your grain.
Silos are cylinders because it is a shape that can be unloaded using gravity and holds up well to the pressures of a filled container.
The height and diameter are largely determined by the cost of land.
Silos that are half the height of the radius of the bin are the most durable and cheapest to build.
Compressed gases are also stored and transported in cylinders because of the strength of the shape.
Here’s a video that demonstrates the concept of volume determined by diameter with popcorn.
Thanks to Margaret Waterman, Southeast Missouri State University, Teacher workshops in Kerala, India, October 2008 (popcorn experiments) for inspiring this activity.
Thanks to Northern Illinois University for their explanation.
Carol Covin, Granny-Guru