If you’re reading this and you’re a mathematician you may not think this is very profound. This is just a little idea I had as I was thinking about the application of the unit circle before moving on with my students to graphing sine waves and other trig functions.

Say you’re on a Ferris Wheel at the county fair, and let’s superimpose the Ferris Wheel onto the unit circle. It’s easy to know when you’re all the way at the top of the Ferris Wheel () and when you’re all the way at the bottom (). Halfway is also pretty easy because it’s when you’re all the way to the sides( and ).

Picture a Ferris Wheel car that’s at 0 degrees. That means that it is exactly halfway up. If the Ferris Wheel spins counter-clockwise, like the unit circle, then the car is also exactly halfway up at 180 degrees.

Now for the difficult question: at what angle is 3/4 of the way up?

It’s easy to assume that you’d be 3/4 of the way up at 45 degrees. This makes sense because 90 degrees is all the way up, and halfway to 90 is 45. But remember: when you’re at 0 degrees you’re only going up, you have absolutely no horizontal motion. This means that for a while you’re going up a lot faster than you’re going to the side. Look at the unit circle and see that , which means that you’d be 3/4 of the way up the Ferris Wheel only when you’ve reached 30 degrees; 15 degrees less than expected!

The other angle with a value of positive is 150 degrees. If we assume that the Ferris wheel is spinning at a constant angular momentum then you’re going to be at the top 1/4 of the ride for 1/3 of the duration of the ride.

Maybe you did get you’re money’s worth (until you realize that you’re also at the bottom 1/4 of the ride for 1/3 of the ride).

Can you think of any other applications of the Unit Circle (without getting into graphing sine waves or other trig functions)?

I assume you used a piece of string with a tack at one end stuck in a paper and a pencil at the other end to draw a circle.

Consider the wheels of a car. How many revolutions must the car wheels turn to go a certain distance? (length)

Consider a circle on a piece of paper. What is the surface area? How much paint would it take to cover the surface? (area)

Consider the amount of air in a balloon. How much air is in a balloon of a certain radius? (volume)

I realize the orbit of the planets follows an ellipse. You may eventually wish to explain the difference between a circle and an ellipse and show how to draw an ellipse.

These are all great ideas, but how can they apply to the unit circle specifically?