Demonstration Highlight: Diameter and Circumference
- Details
- Published: Sunday, 14 March 2021 12:00
Today, Third Month Fourteenth, we celebrate Pi Day. π=3.14…. is a well known fact of science, from our grade school mathematics classes and many dessert-related puns, but what does that actually mean?
Let’s take a look at demonstration A2-11. We see a large metal cylinder mounted on a stand. A bead chain can be wrapped around the cylinder, or pulled off and stretched out straight. When you stretch it out, π is the ratio between its length and the diameter of the cylinder. Technically, π= ½ (c/r), where c is the circumference and r is the radius, radius being the distance from the center to the edge or half the diameter.
Why don’t we define the ratio without the ½? Or why do we use radius rather than diameter? The answer is that using radius makes it easier to generalize to other calculations. If we want to calculate the area, rather than the circumference, we use the square of the radius – which is less annoying to calculate than the square of half the diameter. And even more so when we generalize to three or more dimensions. Ultimately, the factor of 2 falls out from the process of taking derivatives and integrals, just like in elementary calculus.
To see this effect virtually, check out this animation from wikipedia: Pi unrolled. As you can see, if you have a cylinder 1 unit in diameter, its circumference “unrolls” to be approximately 3.14 units long.
Now, check out the Pi(e) Day events at the Maryland Science Center!