To Do and Notice Carefully wrap string around the circumference of your circular object. Cut the string when it is exactly the same length as the circumference. Now take your “string circumference” and stretch it across the diameter of your circular object. Cut as many “string diameters” from your “string circumference” as you can. How many diameters could you cut? Compare your data with that of others. What do you notice?
What’s Going On? This is a hands-on way to divide a circle’s circumference by its diameter. No matter what circle you use, you’ll be able to cut 3 complete diameters and have a small bit of string left over. Estimate what fraction of the diameter this small piece could be (about 1/7). You have “cut pi,” about 3 and 1/7 pieces of string, by determining how many diameters can be cut from the circumference. Tape the 3 + pieces of string onto paper and explain their significance.
Materials cloth tape measures calculators hats with sizes indicated inside them
To Do and Notice Most hat sizes range between 6 and 8. Brainstorm ideas for how such sizes could be generated. Then use measuring tape to measure people’s heads. (As you do this, think of where a hat sits on a head). Use calculators to manipulate measurements. Now compare your results with the sizes written inside the hats. Do your numbers look like they could be hat sizes? (Hint: Try using different units of measurement.)
What’s Going On? Hat sizes must be related to the circumference of the head. The circumference of an adult’s head usually ranges between 21 and 25 inches. The head’s circumference divided by pi gives us the hat size.
To Do and Notice Pick a number sequence that’s special to you—perhaps your birth date.
Go to the Pi-Search Page and type your sequence in the search box at the top of the page. This web site will search the first 200 million digits of pi in a fraction of a second. (See “How it works” on the Pi-Search Page to find out how this is accomplished.) If it finds your sequence, it will tell you at what position in pi your sequence begins and will display your sequence along with surrounding digits.
No result? Try another sequence. The shorter the sequence, the better the odds of finding it.
What’s Going On? Pi is an irrational number, which means that its digits never end and that it doesn’t contain repeating sequences of any length. If Pi-Search didn’t find your sequence of numbers, that’s probably because the sequence occurs somewhere past the first 200 million digits. Note the qualification “probably”: Mathematicians can’t say with absolute certainty that pi contains every possible finite number sequence—but they strongly suspect that this is the case.
As of 2011, pi has been calculated to 10 trillion decimal places. When mathematicians study any sample of this huge number, they find that each digit, 0–9, occurs as often as any other, and that the occurrence of any digit seems unrelated to the preceding digit. This makes pi appear to be statistically random. If this statistical randomness is unending, then pi must contain all finite sequences of digits, including the birth dates of everyone ever born and yet to be born. It would also contain every winning lottery number—too bad we don’t know how to identify them.
Materials large sheet of drawing paper or cardboard meterstick pen toothpicks (30 or more) calculator
To Do and Notice Draw a series of parallel lines on the paper or cardboard, as many as will fit, making sure that the distance between each line is exactly equal to the length of your toothpicks. Now, one by one, randomly toss toothpicks onto the lined paper. Keep tossing until you’re out of toothpicks—or tired of tossing.
It’s time to count. First, remove any toothpicks that missed the paper or poke out beyond the paper’s edge. Then count up the total number of remaining toothpicks. Also count the number of toothpicks that cross one of your lines.
Now use this formula to calculate an approximation of pi: Pi = 2 × (total number of toothpicks) / (number of line-crossing toothpicks)
What’s Going On? This surprising method of calculating pi, known as Buffon’s Needles, was first discovered in the late eighteenth century by French naturalist Count Buffon. Buffon was inspired by a then-popular game of chance that involved tossing a coin onto a tiled floor and betting on whether it would land entirely within one of the tiles.
The proof of why this works involves a bit of meaty math and makes a delightful diversion for those so inclined. (See links at bottom of page.) Increasing the number of tosses improves the approximation, but only to a point. This experimental approach to geometric probability is an example of a Monte Carlo method, in which random sampling of a system yields an approximate solution.
Materials can of three tennis balls cloth tape measure
To Do and Notice Which do you think is greater, the height or the circumference of the can? Measure to find out.
What’s Going On? If you were fooled (and we expect that most people are), blame pi.
You can see that the height of the can is approximately 3 tennis-ball diameters, or h = 3d. But the circumference is pi times the tennis-ball diameter, or c = πd. Pi—3.14—is a little greater than 3, so the circumference of the container is slightly greater than the height