Measuring

1. Measure the distance around a pie with string. Find out how long a piece of string is needed to go all the way around.

2. Measure the distance across the same pie.

3. Divide the distance around the pie by the distance across the pie. That's !

What is ?

is the circumference of a circle divided by its diameter.

is an irrational number. That means that the exact value of can never be written as a rational number (a whole number divided by another whole number). It also means that the digits of never follow a pattern which repeats.

Ancient Egyptian

The Egyptians had an estimate for . They said that the area of an 8"x8" square was about the same as the area of a 9" diameter circle.

Do an experiment to find out if they were right:

relates the area of a circle to its diameter.

1. Measure the area of the circle drawn on the graph paper by figuring out how many squares are shaded. Include your estimate of half and quarter squares in the total.

2. Measure the distance from the center of the circle to the edge in squares; that's the radius.

3. Square the radius (multiply it times itself). Divide the measured area by this number. How close is your result to ?

Probably : Buffon's Needle

Comte de Buffon discovered this result in the 16th century: if you drop a needle on a grid of parallel lines which are spaced the length of the needle apart, the probability that the needle will land on a line is 2/.

Help with our experiment!

1. Drop a bunch of toothpicks on the parallel lines. Write down the number of toothpicks you dropped in the "Number of Drops" column.

2. Count the number which crossed a line and write it in the "Number of Crossings" column.

3. Add your number of drops to the "Cumulative Drops" and write the total on the data sheet.

4. Add your number of crossings to the "Cumulative Crossings" and write the total on the data sheet.

5. Calculate a new value for by dividing the "Cumulative Drops" by the "Cumulative Crossings". and multiplying your result by 2.

1. Figure out the area of the 8"x8" pan by squaring 8 (multiplying it times itself).
2. Figure out the radius of the 9" round pan and square it.
3. Divide the results of step 1. by the results of step 2. This is the ancient Egyptian value for ?
Ancient Greek

The famous Greek mathematician Archimedes came up with a remarkably accurate value for . He decided that the perimeter of a 96-sided regular polygon (a polygon with 96 equal sides and equal angles) was pretty close to the cirumference of a circle of the same diameter. (The polygon's diameter is the distance from a vertex to the opposite vertex). The number he came up with was

= 22/7