The circumference of a circle

Syracuse, a city on the island of Sicily, now a part of Italy. But about 2250 years ago, Syracuse had a Greek king, Hiero the Second. Here he comes. -Archimedes! Long time, no see! -King Hiero, come and sit down! -Where have you been all summer? -Measuring circles. I'm onto something big. -Okay, are you going to run naked through the city again? -Ah, that was a long time ago.

Look at this: Imagine a circle, any circle. -Okay, I'm imagining a... wagon wheel. -Good! Imagine you have a rope around the outer edge of the wheel. -I'm imagining a rope. -Okay, that's enough. Take the rope and put it across the wheel through the center. Then fold it back and forth across the wheel until you’ve reached the end of the rope.

How many times did you fold the rope? -Uh. Two-three times? Is this what you've been thinking about all summer? -Look here: A circle has a circumference. And it has a diameter. A circumference divided by a diameter gives you the number of times the rope fits across the wagon wheel. -Ah...But I was thinking of a huge wagon wheel.

That circle is as small as a pancake. -This is exactly the point! It doesn't matter how big the circle is! The ratio of a diameter to a circumference is the same for all circles. but in that case you just have to take a string and measure one circle, -Okay... and then you're done, right?

And the answer is "a bit more than three." -I've done that. But I want to know exactly. -Of course you do. That's your thing. -Inside the circle I drew a hexagon, like this. -Because I can calculate the perimeter of the hexagon exactly. -Why? Then I doubled the number of sides to twelve and calculated again. So I get a little bit closer to the circle's circumference.

And then I double again... Then I do the same thing outside the circle, too. I haven't reached the circle, but I'm pretty close. -How close are you? -One ninety-six-sided polygon inside and one outside. Now I know the circumference divided by the diameter, to two decimal places. The answer starts with three-point-one-four.

So: there's a bit more than three-point- one-four diameters in one circumference. -Nice! You're right, this will be huge. -What are you going to call this new number? Wait, don't tell me! -You're naming it after me! Hiero-s number! -No. Well, it has to be… something that's easy to remember. -It could be called Ragnhild, like my dog?

The special, useful number that took us a while to discover! Or I know! -I was thinking something short. Like “IT”! -The number that Archimedes was one of the first to calculate so precisely was given a different name. Today we call it Pi. Pi is a letter in the Greek alphabet and it looks like this.

Pi is a number that starts with three, and whose first decimals are indeed one and four. The formula for a circle’s circumference is pi times its diameter. There are ‘pi’ diameters in a circumference. But we can also write the formula as: two pi r .. that is, two times pi times the radius, instead of the diameter.

We can see that the formula for circumference (2 pi r) looks a bit like the formula for area (pi R squared). Don't confuse them. Make sure you're using the correct one for each job. The number Pi is approximately three-point-one-four. But the number has an infinite number of decimals.

No matter how many decimals you take it to, you will never reach the end. 3.141592 653589 793238 462643 383279 502884 197169 399375 ...