Easy as Pi
 

If I asked you what pi is, what would you say?  You might say a special number in math, a constant that occurs in circle formulas, or approximately 3.14.

The easiest way to find pi is to draw a point, take a string and mark all points 1/2 foot away from that point. Then curve the string to follow the circle that you've created.  Stretch that length of string along a measuring tape and notice the measurement. It will be equal to pi. (Remember that the circumference of a circle is 2 times pi times r.)

There is a series of approximation methods called Monte Carlo methods. One of these approximation methods is used to approximate pi. Consider a circle inscribed inside a square of length 2. This makes the circle of radius 1. From area formulas for squares and circles, you should arrive at areas of 4 and pi. In the figure below, the red area is pi, while the red plus the blue area yields 4.

Draw this out on a large piece of cardboard. Take darts and throw them at the "target." As you increase the number of throws, an interesting result occurs. The ratio of the dart hits inside the red compared to any hit on the board (blue or red) gets closer and closer to pi/4.

See if you can explain why. This is not a difficult problem to simulate on a computer or with a graphing calculator program. If you are not familiar with programming, ask a friend to help you create the program. How many throws are necessary to arrive at a good approximation to pi?